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In terms of qualifying for the Arcane Archer your son will need to multiclass. Rangers never gain arcane spellcasting capabilities, so you cannot qualify as a single-class ranger. Typically the Wizard or Sorcerer are used for entering the Arcane Archer prestige class. The fastest way you to qualify from your son's current position is to level up as a wizard. At 7th level he will be a Ranger 5 / Wizard 2, which will give him +6 base attack bonus (+5 from being a 5th level ranger, +1 from being a 2nd level wizard) and the ability to cast 1st level arcane spells as a 2nd level wizard.
Now, there are a few issues that do need to be addressed. The first is that he'll only have the spellcasting power of a 2nd level wizard by the time he's an 8th level character. This is a vast power gulf, and there's no easy way to overcome it. Ordinarily the choice for qualification involves taking many more levels of wizard to better balance your spellcasting and attacking abilities, but given that your son already has 5 levels of Ranger it's too late to do that and his spellcasting abilities are going to be seriously behind the curve for his entire adventuring career.
That leads me into your question of spell level vs caster level. "Level" is without a doubt the most over-used word in all of Pathfinder, and can be one of the most confusing concepts due to this problematic terminology, so let me break down the four different types of "level" relevant to this conversation:
Class Level is your level in a specific class. Your son is currently a 5th level Ranger, therefor his class level is 5. When he next levels up he could choose to become a 6th level Ranger, or he could choose to multiclass and become a 1st level Wizard. This would mean he'd have a class level of 5 in ranger, and 1 in wizard, gaining all the abilities afforded to a 5th level ranger and a 1st level wizard.
Character Level is sum of all your class levels. So if you're a ranger 5 / wizard 1, your character level is 6.
Caster Level is basically like base attack bonus for spellcasters, and determines the potency of their spells. If a spell says its duration is 1 minute/level, then that means 1 minute per caster level. Lots of effects, like spell resistance, concentration, and dispel magic, rely on caster level. Under normal circumstances, your caster level is equal to your class level in the respective spellcasting class. So a Ranger 5 / Wizard 1 has a caster level of 1 as a wizard. However, there are ways to increase your caster level.
Spell Level determines the grade of the spell itself, and which type of spell slot must be expended to prepare and cast it. For instance, magic missile is a 1st level spell, invisibility is a 2nd level spell, and fly is a 3rd level spell. To know what level of spells you can cast, look at the spells per day table for your class. For instance, a 1st level wizard receives three 0th level spell slots and one 1st level spell slot. A 3rd level wizard, on the other hand, receives four 0th spell slots, two 1st level spell slots, and one 2nd level spell slot.
If you have bonus spell slots (usually due to a high intelligence score) just add them to however many you get from the table. However, remember you cannot gain bonus spells for spell levels you do not have access to. A 1st level wizard does not have access to 2nd level spell slots, and thus he can never gain bonus spell slots for that level.
Now that we've covered the terminology, let's cover how this works for the arcane archer. Let's presume your son levels up several times and becomes a Ranger 5 / Wizard 2 / Arcane Archer 3. His character level is 10, and he gains all the abilities afforded to a 5th level ranger, a 2nd level wizard, and a 3rd level Arcane Archer. The Arcane Archer has a special ability and actually increases the spellcasting abilities as a wizard (or any other arcane spellcasting class of your choice, it doesn't have to be wizard). As a 3rd level arcane archer, you add +2 to your wizard spellcasting. This means that when reading the spells per day table you treat yourself as a 4th level wizard, and also have a caster level of 4. Reading the row for 4th level wizards, we can see that he will have access to 2nd level spells.
Now if you're following along, you may notice that having the spellcasting abilities of a 4th level wizard will be very underwhelming when the party is 10th level. Fortunately there are a few feats we can take to improve the matter.
The first is the Additional Traits feat, which can be used to take the Magical Knack trait. This will allow him to treat his wizard caster level as if it were two points higher. So his caster level becomes 6.
The second is the prestigious spellcaster feat. This augments the spellcasting boost of the arcane archer by 1, meaning the wizard now uses the 5th level row for determining which spells he can get. This gives him access to 3rd level spells, which is a big improvement. It also increases his caster level, which (in conjunction with magical knack) means his caster level will be 7.
You can take prestigious spellcaster up to three times. Once when you're a 1st level arcane archer, a second time when you're a 5th level arcane archer, and a third time when you're a 9th level arcane archer.
I hope that helps clear at least some of this up.
Once he becomes that 1st level Arcane Archer are the other classes then "closed off"..? Or can you continue to add levels as you see fit? E.G. he could eventually become level 7 Ranger, lvl 3 Bard, level 2 Arcane Archer, etc?? It would seem like so. Thanks once again.
Every time you level up you can choose which class you want to level up as. A few classes (like arcane archer or paladin) have prerequisites or requirements you must meet, but otherwise you can freely choose every time you level up. Fair warning: this is one of those "you have the freedom to hang yourself" sort of deals. It's very easy to accidentally create a character who can do a lot of things but isn't particularly good at anything.
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A couple notes on Arcane Archer from Ranger - Point Blank Master allows shooting at a target right in front of you without provoking attacks. Assuming that you use Ranger Combat Style: Archery, it's available at Ranger level 6. It's not necessary, but unless a Ranger is built to use backup melee weapons, it's pretty useful.
Choosing spellcaster type and spells is complicated. Make sure to understand how the casting class works, such as what Ability Score it will require to cast spells, and whether armor is a problem. Arcane Armor Training and getting mithral armor helps with spell failure, and the headband items that improve Ability scores help that problem. Low casting power means that spells for self-improvement or utility make more sense than spells for attacking things - Heroism will always work fine on yourself, but Hold Person at low power will rarely hold a strong enemy.
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Archaeologist Bard is a good option for limited Bard levels, since Archaeologist's Luck is easier to use than Bardic Performance.
However, Wizard and Sorcerer can work fine with just Arcane Armor Training and a mithral kikko armor and mithral buckler (you can use a bow and buckler without penalty, and hold the bow in the buckler hand to cast spells). Arcane Armor Training uses a swift action, but I don't think that will matter in this case.
The advantage of Wizard or Sorcerer is that you get more spell power sooner, and some of their class abilities are really useful. The Foresight Subschool of the Divination School for Wizard would be great for an Arcane Archer, even with just one Wizard level.
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If I can make a suggestion, multiclassing into wizard or sorcerer is probably not a good idea. Arcane casting suffers from a failure chance when wearing armor. I don't think our Ranger is going to want to give up armor for magic.
Armor isn't really a big deal given that he's an archer. Because his dexterity score should be his highest ability score by a significant margin, he's already going to be using incredibly light armor. As he continues to invest in dexterity, eventually he won't be able to make use of anything heavier than a haramaki anyways. In the interim, mage armor is more than sufficient (it's the equivalent of +2 leather armor; a wand of mage armor costs 750 gp, and is way cheaper than paying for magic armor, and with 50 charges it will last you an incredibly long time).
If he's got at least 13 intelligence he should be going wizard just because it gets access to better spells earlier than other classes do. When you're behind to begin with, you want to take the option that's a bit accelerated. However, if his intelligence is too low or he's got a big charisma score then the sorcerer is a great pick as well. The only catch is if he's got a low score in both intelligence and charisma. In that case, the Empyreal Sorcerer is the only way to get an arcane spellcaster who uses wisdom. It's not a particularly good sorcerer bloodline, though, so you should only take if you absolutely need to use wisdom as your spellcasting stat. So long as you have at least 13 in your spellcasting stat, you can make it work and just use magical headbands to increase your ability score.