For some reason, I was downloading something at 300kb/s. Then I turned my VPN on and the download speed jumped up to 1.3mb/s.
Why is this? Is it because the VPN server reduces the jumps between me and the server?
There are a few possibilities - unfortunately, the number of hops is irrelevant.
The first is compression - if the data you were downloading is uncompressed, and your VPN offers compression then this could explain it - however most files transferred are likely to be compressed, so this is not as likely as it would seem at first blush.
The second and third options are related, and have to do with your ISP's connectivity and restrictions. Your VPN has found a faster path to the destination data then directly - which could be because -
The ISP has multiple connections, and the direct connection to the data is constrained. The VPN goes across a different connection, which in turn has better connectivity to the source of the data you are pulling, thus you are routing round the congestion.
The ISP is shaping certain kinds of traffic - possibly by type or destination or both - it could even be by content/payload - but that is less likely. By using a VPN, your traffic is being given priority or not being capped, so you are getting better speed.
There are some other possibilities, but these are again less likely - it could be that the VPN is using UDP while your download would typically use TCP, and different optimisations (MTU for example) are allowing better use of your connection. Again, this is possible, but unlikely - chiefly because you would expect either a much smaller or much greater difference in speed.