My DM wants to know how (flavor-wise) my familiar does the help action. My familiar is a small animal (usually spider) hanging on my shoulder while I engage in melee combat. All types of flavor from fun and dumb to serious and edgy is really appreciated.
In general, the familiar can use the help action in combat
As explained in this Q&A, while the familiar cannot attack itself, it can take other actions, including the help action (page 192 PHB):
Alternatively, you can aid a friendly creature in attacking a creature within 5 feet of you. You feint, distract the target, or in some other way team up to make your ally’s attack more effective. If your ally attacks the target before your next turn, the first attack roll is made with advantage.
There is no explicit demand that you need to have strong narrative support, so strictly by the rules, such flavor is not required.
However, as you can see from the answers in the linked Q&A, when you use the help acction it is quite common that the DM asks for an in-game narrative justification for how the help is achieved to allow it. While the Help action is for combat, the rules for Working Together (PHB p. 175) outside of combat for example demand that:
A character can only provide help if the task is one that he or she could attempt alone
Many DMs (mine included) ban help in actions that the familiar may not be capable in by themselves. In this sense, it is already nice to allow the familiar helping with the attack action, when it cannot do so itself. (Although other than the Working Together rule, the Help action makes no such demand). In this case, if you cannot come up with a good explanation how the familiar sitting on your shoulder can feint an attack or otherwise distract the opponent, they may possibly rule that the familiar is unable to grant advantage. There are several options for you
You pick a familiar form that is more able to distract. While a spider sitting on your shoulder may have a more difficult time to do so, a flying familiar getting into the opponents face has a very simple way to distract them. The owl is the best option here, as it has flyby and does not provoke attacks of opportunity when getting away afterwards. The spell allows you to freely exchange the familiars form, for a cost of 10 gp in recasting it.
For a spider, it is more difficult to come up with a justification. Maybe it is a spitting spider (although the mechanics of the monster entry do not say so); maybe it feints jumping or scuttling onto the opponent, maybe the spider scuttling around your neck is distracting to the opponent. Maybe is is coating your blade with glistening spider poison. Maybe it is shooting out web strands that irritate the oppponent. Or maybe the spider is looking for and telephatically alerting you about weak spots in the opponent's defense as Kirt suggests. Or maybe none of that is convincing to your DM, and they will rule you cannot grind advantage for melee attacks out of them.
In some other way
The Help action says (emphasis mine):
you can aid a friendly creature in attacking a creature within 5 feet of you. You feint, distract the target, or in some other way team up to make your ally's attack more effective. If your ally attacks the target before your next turn, the first attack roll is made with advantage.
Listing "All types of flavor from fun and dumb to serious and edgy" that your familiar might use to "feint [or] distract the target" would be, as SeriousBri comments, an exercise in idea generation and something we don't do here. However, if your Tiny (not Small) spider is going to remain on your shoulder during the combat and not (say) jump on your opponent's face, its possibilities for Helping you are rather limited and worth a RAW review.
Looking at the find familiar spell, we can see that:
the familiar has the statistics [including the senses] of the chosen form, though it is a celestial, fey, or fiend (your choice) instead of a beast...Your familiar acts independently of you, but it always obeys your commands. In combat, it rolls its own initiative and acts on its own turn. A familiar can't attack, but it can take other actions [including the Help action] as normal...While your familiar is within 100 feet of you, you can communicate with it telepathically.
If the familiar is going to remain physically attached to you, the most relevant way that it can Help you is by using its own senses, independently of yours, and sharing that information with you instantly and telepathically. In Combat, there are innumerable demands on one's attention, since not only are you watching the opponent you intend to attack, but you must also (PHB Chapter 7, "Hiding" sidebar):
stay alert for signs of danger all around
Narratively, your character is having the familiar watch for openings in your target's defenses when you are looking elsewhere, getting ready to jump out of the way when the enemy mage targets you with their Dex-save spell. Alternately, while you focus on your opponent, waiting for the perfect time to strike, your familiar sits on your shoulder and uses its eight beady little eyes to make sure that no one is coming up behind you. While you yourself are not seeing behind you, you have a trusted partner who will warn you if and when you need to spin around. Because of this, you are able to exploit much more short-lived openings in your opponent's defenses, resulting in advantage on your next attack.