the transition chord is in the key of the new key. that's why it can effectively lead u into the new key.
this is a certain chord progression called the "perfect cadence". a perfect cadence is when you have a 5-1 progression. e.g, B-E, or G-C.
1 is the new key, so 5 will be the transition chord. e.g, if E is the new key, then E is 1, and the transition chord will be 5 notes up, that is, B, or B7 or B9 etc. similarly, if Bflat is the new key, then the transition chord will be 5 notes up: F, and so forth.
according to classical theory of music, perfect cadences are theoretically, the "nice" or "proper" way to resolve a chord progression. usually chord progressions end on the chord 1 of the key. e.g: if the key is in E, the last chord of the chorus or verse(less often) usually will be an E. and the chord before ending on the E will also usually be a B, or B7. go and listen to some songs, those with more standard chord progressions, such as chinese pop, u will notice this(though there are also exceptions when the arranger chose not to adopt it)
applying this concept, this is how the wheel, or circle of fifths or whatever u want to call it, was derived(or at least that's how i think it was derived).
BEADGC -
1) B-E is a 5-1 progression where B is 5 and E is 1.
2) E-A is also a 5-1 progression, where E is now 5, and A is the new 1.
3) A-D; where A now becomes the 5, and D is the new key, the 1.
and so forth.
do u understand better now?