I got confused by using both bis and zu there,
Why it's "bis zum nächsten Mal"?
Some prepositions are used in combination with others. The construction bis zu is used with Dativ (3rd grammar case).
Bis zum (= contraction of zu and dem) nächsten Mal.
There are examples for combined prepositions in English too:
up to, as far as, due to, as for, ...
Why it's "bis zum nächsten Mal"?
As mentioned by splattne some prepositions are used in combination and zum is a contraction of zu dem. Additionally it's worth to mention that:
If two prepositions are used in combination before a substantive or pronoun, the second one determines the case.
2) Wenn zwei Präpositionen hintereinander vor einem Substantiv oder Pronomen stehen, so regiert nur die zweite den Kasus der Bezugswörter:
bis zu ihrem Zimmer, abseits von den Besuchern, je nach dem Angebot, zu je zehn Stück
http://www.linse.uni-due.de/linse/esel/pdf/praepositionen_verwendung.pdf
Bis zum nächsten Mal is a fixed expression like "meiner Meinung nach", which means "according to my opinion"... the meaning of "bis zum nächsten Mal" is "see you next time".
Some sentences are fixed in German and in those sentences two prepositions are used together...