Questions of the form 'What do philosophers think about x?', where x is a broad topic, are doubly difficult to answer comprehensively, because there are so many interpretations of x and so many philosophical opinions on each interpretation. Given that, I will confine my answer to some observations I find interesting and hope that you will too.
The original formulation of quantum mechanics, known loosely now as the Copenhagen interpretation, associated mathematical functions- called wave functions- with particles and systems of particles, and assumed that under certain circumstances wave functions would change instantly when a 'measurement' was performed on the particle or system associated with the wave function. This presented two obvious objections that are still not resolved to everybody's satisfaction today, namely:
How and why can the assumed change happen instantly everywhere?
More importantly, what makes a 'measurement' special, given that it is presumably just an interaction between the quantum particle being measures and whatever collection of particles forms the measuring device?
There have been many different approaches to resolving these objections, one of which has evolved into a range of ideas now known as MWI. I say a range of ideas because there isn't an authoritative single definition of MWI, and some of its incarnations are far more plausible than others (IMHO). The 'pop science' idea of there being literally multiple Universes each as real as the one you suppose you are in is a classic example of an extraordinary claim utterly lacking in extraordinary evidence, and I follow Hitchins' maxim in relation to it- what can be claimed without evidence can be dismissed without evidence.