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The secondary arms feature lists a bunch of things your secondary arms can do, but not what they can't. Does this mean that they can do and qualify for anything that requires a free hand or not?
The secondary arms can manipulate an object, open or close a door or container, pick up or set down a Tiny object, or wield a weapon that has the light property.
Since you've been given a list of things the extra arms can do, the implication is that they can do only those things. That means they don't qualify for everything that requires a free hand, but they may qualify for some things that do, when those activities fall within the realm of what the secondary arms specifically can do (such as manipulating objects). The DM will have to make decisions on a case by case basis for anything that isn't specifically listed.
Grappling isn't on the list of things the secondary arms can do, so having a free secondary arm isn't going to let you start a grapple. The DM might allow it, but since grappling is basically a function of strength, this seems unlikely.
But on the other hand, the Phantom Rogue from Tasha's Cauldron of Everything has an ability called "Tokens of the Departed" that says "as a reaction [...] you can open your free hand and cause a Tiny trinket to appear there" and there seems to be no reason a Thri-kreen's free secondary-arm hand couldn't do this.
Similarly, the rule doesn't directly mention somatic and material components for spells, but if the secondary arms can manipulate objects and attack with light weapons, it seems like they ought to be able to perform magical gestures or manipulate a wand (or a similar small focus item, though probably not a staff), so your DM will have to decide how that's going to work.
Now all that said, your secondary arms can hold objects provided they're tiny (which probably includes most weapons), and transferring an object from one hand to another should not require any specific resource (it's not even a free object interaction to move a sword from right to left, for example, it's just not a thing at all), so you could probably make an attack, then move the weapon to your secondary arms and grapple with the hand that just attacked. This kind of item-juggling would likely allow you to bypass most limitations on which hands can do what even if the DM is being hard-nosed about the Secondary Arms doing only the specific things listed. However, your DM may rule that this is cheese and not allowed, if they are particularly devoted to doing the bookkeeping about what's in everyone's hands.
Personally, I've never found much value in keeping a very close track of exactly what objects characters are holding on a turn-by-turn basis, as long as they aren't being ridiculous about it and trying to duel-wield greatswords, or claim they can wield a shield and a heavy weapon whilst drinking a potion at the same time (or more often, use a mace and shield and cast spells all at once).
