Cambiar and the English for the same, change, both come from the same root: cambiare, Latin also meaning change.
Although this may not be obvious at first, we can see the mapping in the c-m-b of cambiar and the ch-n-g of change. The -m- and -n- are often interchanged; and the -g- and -b- both have that soft sound where you can hear how one can easily turn into the other, although it is a bit less common.
Why did the c- of the Latin turn into the ch- in change? Oh, easy: because it came to English via the French! And French has it own sets of patterns of course!