It's possible to compress videos (with a smaller screen size and/or a loss in quality) but it's far easier to use a file transfer service. There are lots of different ones but they usually allow you to transfer files of up to, say, 2Gb for free but require payment for larger file sizes.
MyAirBridge.com offers a whacking 20Gb maximum size for free though. You simply upload your file and get a link to it. You then email that link to the recipient, so that he can download it. (The link expires after a couple of days though, so he needs to be fairly quick):
https://www.myairbridge.com/en/
Alternatively you can do what I do and open a free email account with GMX:
http://www.gmx.co.uk/
(You don't have to use it for email though, if you don't want to!)
That gives you 2Gb of free online storage space which you can use, for example, to back up your files. However you can also choose to make any of those files 'public' by sending a link to them to anyone that you want to share a file with.