some things to consider:
we often think of numbers as standing for: some amount of something. this is all very well and good until we try to solve: x + 5 = 3. using our usual techniques, we have that x = 3 - 5.
of course, this is is -2, but what does "-2 apples" mean? one way to resolve this, is to think of numbers lying on a line: positive is "one way", negative is "the other way".
this shows that sometimes it is useful to think of "numbers" having two aspects: a "size" and a "direction".
but the space we live in, is not 1-dimensional, it is 3-dimensional. given this, how can we profitably extend our idea of "number" to deal with "3-dimensionality"? one way is to specify "three directions", such as left/right (x-axis), out/in (y-axis) and up/down (z-axis), and give a "size" (coordinate) in each direction. in this way, we replace the number line with the cartesian product of 3 such lines, and get a vector space.
what kinds of things can we represent by such "vectors"? position, speed and acceleration (force), for starters. this alone lets us use vectors to "organize" modelling of physical problems. but it turns out that we can do even more. the x,y and z-coordinates are independent of each other, so any time we have 3 independent variables, we can use the real 3-space to view a problem "geometrically". in fact, there is no reason to stop at just 3, we can consider systems with any number of independent variables.
sometimes, these come as physical problems. here is a really simple example: a farmer tallies his livestock, which consists of ducks and cows, by counting heads and feet. he counts 7 heads and 10 feet. how many ducks, and how many cows does he have? now, one can solve this easy problem just by guessing some numbers. but a systematic way of doing it creates a "head/feet" vector space, and we get the matrix equation:
[1 1][x]....[ 7]
[2 4][y] = 
less contrived situations can easily be produced from economics, where we have a certain number of variables, and a certain number of constraints. in other words, vector spaces don't have to be just "physical spaces", they can be compact ways to keep track of "several things at once". this is, for example, what we do with polynomials. the symbol "x" in the polynomial:
ax2 + bx + c
is just a "placeholder", we might mean the same polynomial when we write:
at2 + bt + c
what we "keep track of" is the vector of "coefficients", (a,b,c). this is not so different than the arabic numeral system we use for arithmetic, instead of "the ones place", "the tens' place", "the hundreds' place", etc., we have "the ones' place", "the x's place", "the x2's place", etc.
in arithmetic, there is "interaction" between the digits, as when we "carry the one". with vectors, we just "remove" the interaction, each coordinate is in it's "own dimension". but many of the same laws of ordinary arithmetic (particular those having to do with addition and subtraction) still hold. so, in some ways, vectors are easier to deal with than numerals, we keep the x's, y's and z's (or more...) "separate".
the point i am trying to make, is that although "vector spaces" may seem, at first, a bit strange, they are very natural, and the rules that govern them are, in a certain sense, "intuitive".