The opening Prologue of "Romeo and Juliet" summarizes the play for Shakespeare's audience:
Two households both alike in dignity/In fair Verona, where we lay our scene,/From ancient grudge break to new mutiny,/Where civil blood makes civil hands unclean/From forth the fatal loins of these two foes/A pair of star-crossed lovers take their life....
So, the enemity between the Montague and Capulet families is absolutely pivotal to the entire plot of the play as noted in the previous post. As well as presenting the hatred of the families, this fight scene suggests the theme of "star-crossed," or fated, lovers. For, with a hatred so intense that a new "mutiny" breaks out in the very first scene, this fight presages the predetermined fate of the love of Romeo Montague for Juliet Capulet.
In addition to its purpose of presenting the theme of fate, Shakespeare often opened plays with fights and scenes like the first one in "Romeo and Juliet" in which there is the ribald gesture and language in the vernacular that appealed to the groundlings, as opposed to the poetic verse enjoyed by the more educated audience. In Scene 1 of Act I there are references to "colliers"/coal dealers and the dialogue is between the servants, people to whom the lower class groundlings can relate. Also, the use of puns such as that upon "colliers, choler, collar" was very popular with audiences of the lower class. Once Shakespeare grabbed the attention of the groundlings with the fight scene and the puns, then, it was easier for the actors to perform and recite the poetic verses, as one can imagine.
This fight is meant to give us an important impression about the Capulets and the Montagues. Their hatred is going to drive the events of the play and we are shown in the very first scene how deep this hatred is. In other words, the impression it creates is that these are two families that hate each other so much (and for so little reason) that even their servants go around trying to pick fights with one another.
The biting of the thumb is a very insulting gesture in that time and place. In the US, it would be the equivalent of showing someone your middle finger. So of course it caused a fight.