Part of it is spreading out the costs over time. None of the individual construction projects is taking 40 years, LA Metro just has a grand plan for the next 15 years, the 15 years after that, etc. Some things take time because of environmental reviews, buying land (or legally forcing people to leave), planning, things like that.

Another reason why they are building in stages is that some of the new transit projects aren't as useful until other ones are finished. For example, there's plans for a big tunnel (a few miles longer than the one you linked) that they're trying to finish in time for the 2028 Olympics (although construction hasn't started yet). A lot of the projects on the other side of the tunnel are being done with the purpose of getting people to the tunnel,