Leather is supple because the fibres can slide over one another. They do this because they are lubricated by fats and oils. When the oils dry out, they can no longer slide and the leather becomes stiff. The fibres themselves are not that flexible so if they are too locked in place and you bend the leather it cracks.
There are many different oils which will do the job, all slightly differently as they also do other things better/less well too.
The first step is to get it really clean, without damaging the leather. This means not flexing it as you clean. The dirt is usually fairly near the surface so you don't want to soak it further in, but that said if there are stains which have gone deep it won't hurt to saturate it, once. Soft soap in water is pretty good, as is isopropyl alcohol (as found in hand sanitiser), followed by a water rinse until clean (but make sure it is dirt not dye which is coming off). Both will remove some oils so applying glycerine (available from Tesco or similar) while the surface is still damp is a good way to reduce the risk of cracking. After that, which oil is largely a matter of choice. Neatsfoot has a particular smell and darkens the leather. A mix of beeswax and lanolin (available from your local chemist or from Ebay) leaves no visible effect and is more waterproofing, but doesn't penetrate so far. Mink oil gives the best flexibility and doesn't affect the colour, but is not very waterproof so a surface treatment would then be advisable.
Avoid anything with silicone in it - it will eventually degrade and stop working, and is a nightmare to get back out.
Alec