EDIT: As this is still the accepted answer, I've made a number of changes to the below text to clarify that, while the "why" is still "probably for multi-factor authentication", this method is not trustworthy.
A phone number enables an easy way to set up two-factor authentication for things like password resets or other sensitive actions. By calling you or sending a text message, the service can confirm that [access to] the phone [number] is a thing you have. An email account is really just a thing you know, the same as a password, because you can access it without anything other than a password (and the identifier - in this case, the address itself - but identifiers are generally considered public). Thus, a two-factor password reset flow might go something like this:
- Click on "Forgot my password" at login.
- Enter your email address, and the site sends you a password reset link.
- Open the password reset link (which contains a secret token), thus proving that the password to your email account is something you know.
- Service sends you a text message with a short secret code in it (possibly after making you confirm your phone number).
- You enter the code from the message, thus proving that your phone is a thing you have.
- Two-factor authentication being completed, you can now set a new password, log into the site, etc.
The advantage to the service provider is that they are now much less likely to have to spend customer service time and possible customer goodwill walking the line between "I can't remember my password" and "somebody hacked my account". That kind of thing can be a significant cost sink, and if the site contains anything sensitive then it may lead to reputational harm to the site if they aren't paranoid enough.
Unfortunately, hijacking mobile numbers is not that hard to do. Even leaving aside technical attacks (faked cell towers or spoofing the phone or passively sniffing the radio traffic to catch the SMS that gets sent), phone companies have demonstrated that they are not trustworthy when it comes to authenticating people correctly before issuing them a new SIM with your number. As such, SMS or call-based second factors should be considered a weak additional protection, and should definitely not be trusted for single-factor authentication.
Second-worst thing would be to use it to send you unsolicited calls or messages, but I don't think I've ever seen this happen (do check for a "use your contact information for marketing or promotions..." option and make sure it's not selected).
In theory, the phone number might be useful to identify you personally (through a directory service / social network / phone book / whatever) and it's a lot easier to create throw-away email addresses than throw-away phone numbers. It's unlikely, though. If you're the kind of person who uses a fake name and a single-use email address when signing up for something, then putting your real phone number in there is obviously a risk. Otherwise, it's probably pretty harmless.