How can I improve or work a resource that is more than three tiles from land?

Notice there's a difference between improving a tile and working a tile. Usually you will improve tiles around your cities and then send citizens to work these tiles, but this connection is not necessary.

Improving a tile

To improve a tile, you need it to be within your cultural borders (the tinted area) and reachable by a worker/work boat, and you need to have the appropriate technology. Once a tile is improved by the worker, you get any luxury or strategic resource on it, but that doesn't mean you get the tile benefits (e.g. +production, the small hammers) - you have to work a tile for that.

Pay attention: cultural borders can reach practically everywhere, since cities can expand these borders up to 5 tiles and great artists can expand it arbitrarily further. I once used 3 great artists to reach a far away tile (Aluminum...), it takes time because there's a cooldown between cultural bombs, but it's doable. "Culture bomb" is the name of the great artist's ability which does it. However, in practical terms, if you want access to any resource the best way is to build a city up to 3 tiles from it. Usually when I build a new city I only consider things in radius 3 to be interesting, anything beyond it just usually takes a lot of turns to get.

Notice that some worker actions, such as cutting down a forest or building a road, do not require the tile to be within your cultural borders.

Finally, if you build a city directly on a tile it doesn't count as improving the tile (the tile benefits won't change) but you do get whatever resource is on it.

And as Raven mentioned, to improve tiles at sea you must construct a work boat from a coastal city. Unlike workers, work boats are consumed when they construct an improvement, but they are otherwise very cheap.

Working a tile

To work a tile, you need it to be within 3 tiles of one of your cities, and then that city gets the benefits listed on the tile. In your screenshot, Krakatoa is unworkable, and no civilization will never get the benefits listed on the tile. Notice that even tiles in range 3 are not automatically worked, a citizen must be allocated to it. You can manually allocate citizens from the city screen but the computer takes care of it by itself, if you're new to the game you don't need to worry about it at this stage.

Your puppet cities also send their citizens to work nearby tiles, but you cannot change that allocation.

A tile doesn't need to be improved in order to work it, but most tiles benefit from improvement so ultimately it pays to improve any tile within range 3 of a city. Improving a tile beyond range 3 which doesn't have a resource on it offers no benefits (unless you improve it to a fort or something).

In terms of "working" a tile, all it means is to have a citizen from a city gathering resources from that tile.

In terms of improving sea-based resources, you need to instruct a coastal city to build the "Work Boat" unit, which is unlocked with sailing, afterwhich you can move them onto an ocean resource and "create fishing boats" (which will consume the unit).

EDIT: You could get the marble, just build a city on it. However, unless you really need the happiness it isn't going to be a super stellar city. Marble's benefit of cutting wonder construction time only benefits the city that is working it. Since the one land tile city's production is going to be low, it isn't going to be a great wonder builder.

I don't think there is a way for you to benefit from Krakatoa beyond the free 1 happiness for finding it.

Further EDIT: You could also build a city on the shore. Max expansion for a city is five tiles, you can only work the first three, but luxury resources inside of your borders (that are enhanced [plantation, quarry, etc.]) still provide the happiness bonus.

This is an example of a tile being worked, in this particular example that tile provides Wine (+1 food, +1 production, +4 Gold). You should see something similar on your natural wonder.