Why is the Mersenne Twister regarded as good?
The initial Mersenne-Twister (MT) was regarded as good for some years, until it was found out to be pretty bad with the more advanced TestU01 BigCrush tests and better PRNGs.
This page lists the Mersenne-Twister features in detail:
Positive Qualities

Produces 32-bit or 64-bit numbers (thus usable as source of random bits)
Passes most statistical tests

Neutral Qualities

Inordinately huge period of $2^{19937} - 1$
623-dimensionally equidistributed
Period can be partitioned to emulate multiple streams

Negative Qualities

Fails some statistical tests, with as few as 45,000 numbers. Fails LinearComp Test of the TestU01 Crush and BigCrush batteries.
Predictable — after 624 outputs, we can completely predict its output.
Generator state occupies 2504 bytes of RAM — in contrast, an extremely usable generator with a huger-than-anyone-can-ever-use period can fit in 8 bytes of RAM.
Not particularly fast.
Not particularly space efficient. The generator uses 20000 bits to store its internal state (20032 bits on 64-bit machines), but has a period of only $2^{19937}$, a factor of 263 (or 295) fewer than an ideal generator of the same size.
Uneven in its output; the generator can get into “bad states” that are slow to recover from.
Seedings that only differ slightly take a long time to diverge from each other; seeding must be done carefully to avoid bad states.
While jump-ahead is possible, algorithms to do so are slow to compute (i.e., require several seconds) and rarely provided by implementations.

Summary: Mersenne Twister is not good enough anymore, but most applications and libraries are not there yet.