Question in the title: how does temperature affect conductivity?
Conduction is the travelling of electrons across the system. A perfect monocrystal is a perfectly regular grid, and the solutions for the electrons are the Bloch waves. Electrons behave like waves travelling through the solid. Any interference will slow down this waves; electrons will collide and get scattered, thus becoming slower. To picture this: each drunk person's steps move in different directions, one forward, one a bit to the right, other slightly backwards... so they go slowly.
Temperature makes the solid to vibrate. This vibrations are called phonons (not photons) and are seen by electrons as disturbances on the grid. When a travelling electron hits one, the space is suddenly not so regular anymore, and gets slightly deviated (scattered), and thus, slowed down.
In semiconductors there is an extra thing, as we will soon see.
The other question: how does RAM keep memory after being off?
A computer memory is a bunch of 1 and 0. We can codify this information in a circuit by holding electrical charge or not. The actual implementation uses transistors and semiconductor physics, but seen from afar, it is similar to a capacitor. When we want to store a 1, we charge it up, and when we want a 0, we empty if of charge.
This system can be done very fast, so we can read and write data quickly when we are doing computations, but has problems. Our capacitor is far from perfect, it leaks charge. So, if a 1 means that the capacitor has a voltage difference of $1 V$, if we leave it alone, charge will slowly flow, and the voltage will drop. To prevent memory from degrading, RAM reads continuously the data. Say a certain cell a while after being written reads $0.7 V$: it was probably a 1, so the RAM will erase it and rewrite a brand new $1 V$. This process took less than $100 ns$.
When we power off the computer, the refreshing stops, and the charge of each cell quickly degrades. After some time, enough electrons have travelled from one side to the other so that nobody can tell if there used to be a 1 or a 0 on that spot.
To understand the temperature effect here we need to dig a bit deeper. At the very core of these chips, there are two semiconductors in contact forming a p-n junction. They are usually two pieces of silicon with small portions of two other materials, one for each side.
In solid materials, electrons at rest are in the valence band, a permitted region of energies that is full of electrons, so it doesn't allow movement: they are still. There is another layer, with more energy, called the conduction band. This is empty, so electrons can move. In a conductor there is no energy jump between these two, electrons in the valence band can just be in the conduction band and move along. In insulators, there is a large gap: we need a lot of energy to make the lazy electrons move to the level where they can move (this is why screwdrivers specify a voltage limit, more than that will give enough energy to the electrons of the handle to move and fry you). Semiconductors are somewhere in the middle: they need some energy, but not much.
This little bit of energy can be obtained through several means, that give rise to all the richness and possibilities of electronics. But it can also be obtained by heat: an electron can just be hot enough so it has the right energy to jump and move. The hotter the material, the more electrons there will be with the right temperature, and thus the easier it will be to conduct. Conversely, if you freeze your memory bank, very few electrons will have enough energy to make the jump, and the charge in your capacitor will remain for longer.
