"Reisen" just means "to travel" and has no element of direction to it.
This is different for "anreisen". It also contains "reisen", so it means travelling, but a certain kind of travelling: travelling with the element of to this place. For instance:
Meine Frau wird morgen anreisen.
means my wife will travel tomorrow to this place (implied: where I am right now), which is why it can mean "to arrive". Where the implied "here" is can be changed by specifying a place:
Er ist am Dienstag in China angereist.
He arrived on Tuesday (not where I am, but) in China.
Notice that there is not only "anreisen" but also "abreisen": to travel with the direction of away from here - to leave. See below.
German in general has this trend to derive verbs by adding prefixes of all sorts to some "base verb". In English this also the case but not as common, for instance "stehen":
stehen (to stand)
anstehen (to stand in line, to wait, also: to be imminent/planned)
verstehen (to understand)
bestehen (to be, to exist, also: to insist)
umstehen (to surround sth.)
gestehen (to confess)
zugestehen (to concede, to grant)
zustehen (to have [earned] a right to sth.)
and so on. This list is not complete, it is just an example of how verbs can be derived from some base verb. The same is true for "reisen":
anreisen (to go to here or some designated place)
abreisen (to leave)
verreisen (to travel [away from where one is normally])
nachreisen (to travel to someone else, who has travelled before)
bereisen (to travel to some place)
ans so on. This list is also not complete, and some words would be worthy of more explanation than the short translations I gave.