We don't really know exactly why the Montague and Capulet families are fighting with each other.  In the Prologue, the Chorus says that they are "Two households, both alike in dignity" (line 1).  The families are of equal status and honor (and wealth, too, we can assume), and they have between them "an ancient grudge" (line 3).  The feud is apparently so old, then, that no one ever even discusses how it originated; they just continue to brawl nonetheless.  The only thing the Capulets and Montagues seem to fight about in the play is family honor: whose master is better, whether or not Romeo, a Montague, should have been in attendance at the Capulet party and whether his presence dishonored the Capulets (according to Tybalt, it does).  None of the characters reveals the original cause of the fighting between the families, and so we never learn it.

Why didn't Romeo and Juliet's families like each other?

As the others have noted, there is no reason but it does highlight how silly it was to have it in the first place. If there wasn't a feud, Romeo and Juliet would have lived happily ever after. Then again, this would have been a very boring play without it.

Why there was a strain between the families of Romeo and Juliet?

The very first mention of this feud is in the play's Prologue.  The speaker refers to both families, the Capulets and the Montagues, as "alike in dignity" (line 1); thus they are of the same class and social standing.  Further, they bear toward one another an "ancient grudge" that, during the course of the play, will result in "new mutiny" (3).  The cause of their long-standing feud is never revealed to the audience; however, we see the new examples of insults the families trade in the play.  In fact, the very first scene presents such a conflict when Sampson and Gregory, of the house of Capulet, end up engaging in a street fight with two servants of the house of Montague when Sampson asserts that his master is "better" than theirs (1.1.61).  Further, when Romeo and his friends go to the party at the Capulets' house, Tybalt takes it as a terrible insult and wants to fight Romeo the next day.  When Romeo refuses to fight, Romeo's friend, Mercutio, fights Tybalt instead, believing that Romeo dishonors himself by resisting.  When Tybalt kills Mercutio, Romeo kills Tybalt, increasing the tensions between the houses even more.

It is possible that, since both families are so well off, that the patriarchs fought over land or property or something of that effect.  Perhaps they wounded each other's pride a long time ago and have never dropped the grudges they held.  We simply don't know for sure because it is never stated in the play.