This is an excellent question which stumped me for some time as a student.
Systolic pressure generation is easy. The heart is a pump, and by contracting and forcing the left ventricle's blood into the systemic circulation, it raises the pressure dramatically. It's a little like raising the pressure in a water balloon by suddenly dumping a whole lot more water inside it.
But then the contraction is over, and the heart relaxes and refills. Its systolic pressure generation is intermittent, almost as if you were turning a pump on and off and on and off and on again. During the 'off' periods (i.e. diastole), the pressure should therefore fall to zero! Yet clearly it does not. When you go to your doctor and measure your blood pressure, you might find that it is somewhere in the region of 130/80. Far from plummeting to nothing, the pressure in your brachial artery alone during the 'off' period of the pump is enough to raise a column of mercury 80 millimetres high into the sky! How the hell is this diastolic pressure generated?
The first key to the answer is to note the arterioles (that the arteries feed into) are a source of resistance to blood flow. They are fundamentally a large number of narrow pipes that the rapidly-flowing arterial blood has to force its way into, and they therefore impede the blood's passage.
The second key to the answer is that the arteries are elastic; that is, they can distend if the pressure in them rises. And the pressure in them does rise - on the one end they have heart tirelessly pumping blood into them, but on the other side there is the arterioles' resistance to allowing blood to leave them. This mismatch between the amount of blood that enters vs exits the arteries during systole means that the pressure within them rises. And, because they are elastic, they bulge outwards. (Again, you might find the water balloon analogy helpful.)
Incidentally, this 'bulge' is what you are feeling when you palpate a patient's pulses. It is fundamentally caused by the systolic rise in pressure forcing the elastic arteries to bulge outwards a little. This bulge travels down the arteries rather rapidly:
(It is worth considering that if your arteries were as rigid as steel pipes, their lack of elasticity would make it impossible to feel a pulse, however well they might allow for the passage of blood.)
The stage is now all set for diastole. The distension of the arteries causes them to 'fight back' and try to squeeze the 'extra' blood out. (This is the same principle that you demonstrate when you stretch an elastic band. The elasticity allows the band to stretch, but the stretch generates a force within the band that 'wants' to make it contract back to normal again.) But try as they might, the 'extra' blood pushed into the arteries by the heart can't all make its way through to the arterioles by the end of systole. The resistance in the arterioles turns out to be enough to slow the arterial blood flow down so much that there is still more of this 'surplus' blood to be expelled come diastole.
And that ongoing process generates diastolic pressure. The distended arteries generate a force, due to their elastic properties, that will tend to propel blood onwards until they are no longer distended anymore. They never quite get there, of course (or else the diastolic pressure would finally reach zero) - systole comes before they can ever accomplish their thankless mission. And thus, we have diastolic pressure.