BitTorrent is a perfectly safe (sometimes quicker) way to download files, I'll compare the two:
HTTP Download
This asks the web server for the file, and your web browser downloads it from that one source. Speed will vary between your connection and server. Bandwidth isn't free neither, so there is a slight cost to the project for all the bytes needed to send the file to you. That's why BitTorrent is preferred where possible.
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Security: It's highly unlikely to happen, but if your connection is involved in a man-in-the-middle attack, the web browser could be downloading a modified version. That's where checksums come into place, which I'll describe further below.
BitTorrent Download
Instead of downloading from one source, your BitTorrent client securely connects to other clients who are sharing the same file. You could be downloading from your next door neighbour for all we know. 
Your client asks others for different pieces of the file you don't have so they can be transferred to you, and vice versa -- others who are looking for a piece of the file that you have but they don't, can have that piece too.
This approach requires you to upload too. Since there can be many sources, this can greatly accelerate the speed of the download. Ubuntu MATE's torrents also have web seeds, so there's always a server to provide the files if nobody is around to seed.
While you are downloading and haven't quite got the full yet, you are considered a leecher.
As soon as you have everything, you become a seeder.
It's a good idea to upload as much as you downloaded so others benefit too - at minimum a 1.0 ratio. Others may be generous keep their clients running under they achieve a 1.5 or 2.0 ratio. Those who don't bother seeding may feel guilty for not giving back as much as what others gave to them. That's how some torrents may end up dead with no seeders and too many leechers.
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Security: BitTorrent verifies data as it's downloaded, so if a piece doesn't match, then it will download it again from another source. Nothing malicious can get through as it is checksummed, not unless it was a dodgy torrent to begin with.
Checksums
The file is either exactly the same or it's different. It's the foolproof way to know if the file is an exact replica or had been modified along the way. The download page has a link to these checksums, and you can use a utility to verify them.
This is also recommended, as the page states:
If you direct download the .iso image please make sure the [appropriate MD5 hash][1] matches.
Under Ubuntu, you can use the md5sum command:
md5sum /home/user/desktop/ubuntu-mate-15.10-desktop-amd64.iso
If the checksum matches, it's a perfect clean copy, otherwise, it's been modified -- sometimes could be caused by failing hardware or a broken download.
[1]: http://cdimage.ubuntu.com/ubuntu-mate/releases/15.10/release/MD5SUMS