Update in reply to your comment: “Then what causes that light, and the spark?”
The ions get accelerated in the strong electric field and collide with one another. This causes heating (typically around 30,000ºC). Material this hot gives off light as well as heat (incandescence). It is worth noting that this sudden heating causes a rapid expansion of the surrounding air which is the cause of thunder.
Also, light (photons) are released when the electrons recombine with the atoms and molecules. A photon is emitted for each recombination. The wavelength (colour) depends on the atom’s energy levels. To find out more read about ‘characteristic radiation’.
Look at diagram: https://girlinclouds.files.wordpress.com/2013/02/lightning-diagram.jpg
Q1: What happens when the negative and positive charges collide during lighting?
Answer: The positive charges are ionised atoms and molecules from particles of water. The negative charges are (usually) electrons. They combine to produce neutral atoms or molecules.
Q2: The negative charges from the clouds and the positive charges from the ground meet, and then they flow in both directions and hit each other, right?
Answer: You are describing a 'ground flash'. The charges flow *before* they meet. As the charges flow, they also hit other atoms/molecules causing heating and further ionisation. Then the charges combine as described in Q1.
In the case of a ‘cloud flash’ the ground charges aren’t involved.
Q3: So what happens then? They hit each other and just disappear?
Answer: As for Q1.
Q4: Why don't they hit each other while they are in the cloud?
Answer: They can do. It is then called a ‘cloud flash’.
Whether you get a ground flash or cloud flash depends in a complicated way on many factors such as height/shape/charge-distribution of the cloud.
Q5: Just because there is an equal number of positive and negative charges in the cloud they don't hit each other? Why?
Answer: They are physically separated because the larger water particles get a negative charge and the small ones get a positive charge (no one fully understands why). The smaller (and therefore lighter) positive particles get pushed towards the top of the cloud by the up-currents. They are kept separated from the heavier negative particles by the effect of the up-currents until the conditions change. Then the charges recombine – either’ silently’ or as lightning.