Once in a city train in Bay area, I saw a plate which says something similar to "federal law requires this seat to be reserved for disabled people". What part of the constitution authorizes the US congress to establish such a law?
The basic constitutional underpinning is the Commerce Clause and the 14th amendment (see Americans With Disabilities Act of 1990, Pub. L. No. 101-336 § 2(b)(4)). As a general rule, if you do a thing that has potential commercial impact involving another state (such as growing feed for your animals and therefore not buying feed from a farmer out of state), then Congress can pass laws that restrict such actions. ADA was not constructed specifically to address seating on trains: that is just a consequence of the law.
It could be that BART uses federal money from Department of Transportation (DOT) allocated to mass transit systems by the Federal Transit Administration (FTA), which is the department that deals with public transportation. By taking the money, the FTA can then enforce federal laws on the system.
This is not a violation of Commerce Clause because the state of California voted to accept the money from FTA. Thus, the State is Choosing to enter into contract with the Federal Government so that the State can get money for their system. All the feds ask in exchange is that they comply with Federal laws, which include handicapped seating.
In addition, BART has several Amtrak Terminals it services. Amtrak is a federal for profit entity that is interstate transit. Thus, BART does participate in Interstate Trade, and can also be regulated by the Federal Government.