If somebody knows what does that mean?
They seem to claim that they only expect you to make web pages on their service, and any kind of application/web service that makes API calls are unlikely to work well on their service.
Could you explain the difference between web host like Amazon Lightsail and others
There are "shared web servers" -- these are the cheapest, and is the kind that the OP initially set up.
These give you a directory to upload your web files to. Those web files might be intended for some particular web development language (PHP, Ruby, Python, etc) and typically use a "one request is one process" model. You receive a HTTP request, you do whatever queries and processing you need to do (if any) and you respond with a web page or some other web resource (image, XML, JSON, ...) You cannot upload code that's not supported by the web server host -- you can't build your own C code for processing network requests -- and you cannot run processes that stay alive for longer times, talking to many players at the same time.
These are okay for "my own photo album" or "jimmy's pizza ordering page." If the page sometimes takes a second or two to load, that's not a big deal. All the traffic goes over HTTP, which is a request/response, bulk transfer protocol on top of TCP, with no particular guarantees around latency. Also, typically there are 1000s of "home directories" on each of these servers, serving a bunch of different, small, web pages all at once. (The server knows which directory to look in based on the URL and Host: header in the HTTP request.)
Then, there are "metal servers" -- a computer is plugged in to the network, and has an OS on it, and you get administrator/root access. It's up to you to make sure the hard disks are partitioned the way you want them, that the software you need to run on there is installed, and that security exploits get patched. You can build whatever software you want and install on these servers. For action games, which almost always use UDP networking, this is great, because you can spin up a game server on a known port, and point game clients at this server/port. Draw-backs are that you need to be much more aware of how to administer servers in a safe, secure, and efficient manner.
In addition to game servers, web sites that are bigger than suitable for a shared web host can use one of these servers (or more) and install their own web server and database server as the software that runs on these servers. This allows you to avoid the "noisy neighbors" problem of shared web hosts.
Because these servers are a full computer that costs a lot of money to buy, server hosting companies will charge significantly more per server instance. Typically a few hundred per month, at least (more if you want fancy memory / RAID disks / multiple CPUs and so on.)
Then, there's an intermediate step. You don't need a big server with 256 GB of RAM and a 10 TB flash RAID array and dual 10 Gbps uplinks to the router. You just need some fraction of that -- one or two cores, one or a few gigabytes of RAM, a dozen gigabytes of disk space for your software, and a terabyte or two of network transfer per month. You're OK with sharing the same physical hardware with others, as long as you get the guaranteed memory and processing power that you pay for. Various "virtual private server" ("VPS") solutions deliver this kind of server. For most intents and purposes, it "looks like" a small metal server (and has some of the same problems of needing security patches, administration, and so forth,) but the cost of actually sticking hardware in a rack in a data center is spread across many people who each share a defined portion of the server. Thus, this is cheaper per month than a full server.
This is what Amazon Lightsail, Linode, Dreamhost VPS, and Interserver end up selling you. Amazon has the best name in the business and the most mature set of infrastructure tools, and thus charges the most. Dreamhost has a business that's more around easy-to-manage web sites, and provide a lot of value-add if you use their software versions, and thus charge extra for that. (They also have shared web hosting.) Linode gives you a little more than Amazon for your money, while having a smaller variety of tools around their images, but are still a well-known actor with some really big customers. And, finally places like Interserver is just "give me a server and get out of the way" bare bones, which lets you get down in price a few more dollars at the bottom end. On the other hand, their data centers and internet uplinks are more oversubscribed than Linode or Amazon -- you get what you pay for.
There are of course many more online server providers of various kinds -- Heroku, 1and1, Peer1, ServerBeach, Rackspace, and the list goes on. The four above are the ones that I have direct experience with that has been good enough to recommend if you need their particular kind of service.
Once you outgrow VPS/cloud solutions and metal-leased servers, you will build your own data centers in some co-location facility, negotiate for bandwidth with upstream providers like Cogent and Level3, and screw your own servers into your own (rented) racks, where you have to build your own in-datacenter network infrastructure (routed or switched? Overprovisioned or nonblocking?) and have to worry about things like power density (130W CPUs perform great, but if you try to cram two of them per motherboard times 42 of them in a rack, you will draw more power than most power distribution and cooling will let you get away with -- are watercooled racks worth it?) Co-location is cool and all (or, if you're facebook/google, you build your entirely own data centers) but very few indie games grow to the point where they need to worry about that. Find a VPS vendor you can trust (linode or amazon) and stay there for as long as you can.