When doing in vitro fertilisation (IVF), the benefit of letting the embryos grow to blastocyst in the laboratory before transferring them to the uterus, is that it enables better selection of the best embryos. If an embryo has made it to this stage, then it has already survived many obstacles, so the chances of it resulting in a viable pregnancy are higher.
If you have a blastocyst transferred as part of your IVF there is a strong chance of becoming pregnant. HFEA website currently quotes 58% chance for someone of my age (<35 years).
For some people it isn’t appropriate to try and do a blastocyst transfer, as the risk of the embryos not surviving to the blastocyst stage is too high. In these cases, embryos may be transferred on day 2 or 3.
HFEA information about blastocyst transfer
CRGH page on blastocyst transfer
This webpage has a picture of a blastocyst