Users have had the option of sharing a video that starts straight from the time they wanted to for a while now, but it's not the most user friendly of features. It's not overly complicated, you just have to add a parameter to the video's URL. There's also a way of making sure embedded videos start at a certain point as well.
Now though, the feature is available in the sharing options section below any YouTube video. If you Click on share, you get a short URL starting with "youtu.be" and containing the video code as the short link.
If you click on Options though, you get a few more features, the ability to share the HD version of a video, if available, to use a long link instead of the shortened one, but also the option of starting the video at a certain point.
"Did you know you can change a link so a video will start at a certain point? Just click Share > Options on a video page or add #t= and the number of minutes and seconds at the end of the URL. For example, if I wanted to start at 4 minutes and 7 seconds, I'd add #t=4m7s and I'm done!," YouTube explained on Google+.
This ability has been around for some four years now, but it wasn't widely known and few people used it.
Nevertheless, it was a useful feature, so having it in the sharing options is a good decision, even though the sharing options aren't all that easy to get to. An alternative is to right-click on the video and choose "Copy video URL at current time."
You can also make sure embedded videos start where you want them to start and it's easy to do so, just modify the <iframe> code provided by YouTube by adding the #t= parameter at the end of the video URL.