It's a familial thing. Kids don't decide; they're born into it.
If you're looking for a more historical look into why the original supporters chose a club, from Wikipedia:
Today there are no evident geographical, political, social or religious divides as in other derbies, although for many years a sectarian divide did exist within the city. It is unclear how (if at all) this influenced the support bases of the two clubs and more recent research has indicated that it was more likely to have been a political allegiance that influenced support.[6] During the 1950s and 1960s Everton became known as the Catholic club mainly as a result of successful Irish players Tommy Eglington, Peter Farrell and Jimmy O'Neill as well as manager Johnny Carey. This in turn caused Liverpool to be thought of as Protestant club, not signing an Irish Catholic until Ronnie Whelan in 1979.[7] However, it should be noted that this notional divide was never seen as a basis for supporting a certain side as is the case with Celtic and Rangers. In truth both teams have strong support from all denominations as well as many fans from Presbyterian North Wales, Northern Ireland and Republic of Ireland. Most importantly, the actual clubs themselves did not act to strengthen sectarian divides and in fact both clubs stem from a Methodist origin
If you're really interested, I'd recommend reading the citations provided because they're pretty interesting.
As for the initial split itself, Wikipedia is also helpful:
Several board members of Everton were members of the Liberal Party who were associated with the National Temperance Federation whilst Houlding was a Conservative Party member and a brewer whose business interests were diametrically opposed to the temperance movement. Politics and disputes over money meant that Houlding was increasingly at odds with other members of the Everton board. The result was that in 1892 the Everton directors vacated Anfield and purchased a new ground at Goodison Park on the other side of Stanley Park. Houlding responded by founding a new club to use Anfield: Liverpool.