Disclaimer: I'm a first-year law student. If anything I say seems to contradict something any objective lawyer (or even a further along law student) has said or will say about this, listen to that person and not me. That said, I'll attempt to clear up some of the things at issue here.
Q: What happened?
A: Adam Silver banned Donald Sterling from the Clippers for life. This means that he can't associate with the team in any way: "As part of the lifetime ban, Mr. Sterling may not attend any NBA games or practices, be present at any Clippers office or facility, or participate in any business or player personnel decisions involving the team. He will also be barred from attending NBA Board of Governors meetings and participating in any other league activity." He also fined Sterling $2.5 million, which is the maximum allowable fine.
Q: Does Donald Sterling need to sell the team?
A: Silver said repeatedly that he will urge the owners to force a sale of the team. The NBA bylaws state that the "interest of any owner may be terminated" by a 3/4 vote of remaining owners (that's 22). As Seth mentioned, it seems like Adam Silver thinks that once the votes have been cast, Sterling is gone for good. However, nothing is that simple when it comes to the law. Silver knows this, but admitting that Sterling probably can't be forced to sell wouldn't be a very smart idea. Also, the courts won't force Sterling to sell the Clippers on their own, so the bylaws need to actually be invoked if this sale were to ever occur. Another thing to note is that this decision only affects Donald Sterling and not his family. I'm not sure how much of the team he owns, and I'm sure it's a majority, but it's possible that even if he sells his interest, his family will still own a hefty percentage of the team.
Q: How will Sterling respond to this?
A: He's going to sue the NBA Board of Governors. Upon a cursory reading of Section 13, I see a few issues that Sterling can attempt to exploit assuming he does bring a lawsuit. The full NBA Constitution isn't publicly available (though I'd bet Deadspin or the like gets their hands on it soon enough), so my opinions aren't fully formed. I'm only working with Sections 12 and 13 here. First: the NBA's options here are to invoke either 13(a) or 13(d) when it takes this vote. The other sections of Section 13 refer specifically to scenarios such as an illegal sale, bankruptcy (which is dealt with extensively in Section 12), and gambling. The success of an attempt to invoke 13(a) or 13(d) depends on what's in the rest of the Constitution. If there's a specific section elsewhere in the document that outlaws bigotry and discrimination, 13(a), which states that the vote can happen if an owner "willfully violates any provisions." Of course, it will need to be determined whether what Sterling said fits the definition of what's in the Constitution. As for 13(d), which mentions adverse actions, it's a bit of a stretch, in my opinion. Sterling doesn't really have a contractual obligation to not be an asshole, so 13(d) probably doesn't apply.
Another issue is the word "terminated." It's likely that "termination of interest" is defined elsewhere in the Constitution to effectively mean that the owner needs to sell his ownership stake, but I can't be sure. Courts aren't too keen on, in effect, stealing property from people. If Sterling is not allowed to sell his stake in the team under the Constitution, that'll be a problem for the courts. To be clear, I'm sure the NBA had very good lawyers draft this document and it's doubtful that Sterling will win, but he'll have legitimate arguments and will be able to hold this decision up in courts for a long time. Sidenote: ESPN Legal Analyst Lester Munson said on SportsCenter that he thinks this will be settled in arbitration. If that's the case, the NBA will get to pick the arbitrator and Sterling will likely lose.
Q: How much power does the Constitution/Bylaws even have over NBA owners?
A: I'm not sure, but I think it has less power than you might expect. The NBA is not exempt from anti-trust laws, which means that the league is in more of a supervisory role than an ownership one. From what I understand, the teams engage in revenue sharing and such as a form of interstate commerce and the NBA basically regulates said commerce. Bylaws are often associated with housing corporations such as condos and coops. The way bylaws usually work in those cases are that any bylaws not transcribed when the association was first formed aren't presumptively reasonable and will only be upheld if they further some legitimate purpose. It's safe to say that denial of discrimination is a legitimate purpose, but either way, there is ambiguity. A court does have the power to say that the NBA can't hold Sterling to this whole document, but I doubt it would actually rule that way.
Q: What recourse do the players have if they don't want to wait this out?
A: Not much. Rumors are floating around that the players will ask to be declared free agents by the NBA. Adam Silver mentioned that he's not considering that at the moment. He's right to say that, because it's highly unlikely that would ever be allowed, legally speaking. As I mentioned, since the NBA isn't exempt from anti-trust laws, the Clippers players work for the Los Angeles Clippers, and not the NBA. Even if the NBA were to find some provision of its Constitution that allows it to free the whole team from its contracts, such a decision would never hold up in court. Sterling would sue the NBA and the NBAPA for such an action, and the court would likely hand down what's called a preliminary injuction, which would force the players to adhere to their contracts until it rules on the matter. If a court were to decide that the players can get out of their contracts, it likely wouldn't make that determination for years, at which point the players wouldn't really care much. I highly doubt the court would ever rule in the players' favor, anyway, as it would set a very dangerous precedent for invalidating contracts. The government could conceivably step in and invalidate the contracts itself based on the some law against hostile work environments, but with Sterling banned, a court would probably say that the environment is no longer hostile.
That's all I have off the top of my head. Again, if you find someone who has more experience and no stake in the outcome of this case contradicting anything I've said, listen to that person. I hope this helped a little bit.