Forward traction is basically the amount of grip your vehicle has when accelerating. This is only useful on wheels that are connected to the engine.
Summary: The vehicle accelerates faster the more forward traction it has

Now, sideways traction is more complicated
It is the traction your vehicle has when turning.
little sideways traction on the rear wheels on a car make it do a thing called "Oversteer", mainly on rear wheel drive vehicles
Summary: Makes your car very drifty

little sideways traction on the front wheels means that the car whill "Understeer", much easier to control than "Oversteer", but much more penalizing in terms of performance

Understeer happens a LOT on Front wheel drive cars (but not in SP, in this game there are practically no disadvantages in putting front wheel drive in your car)
It's much harder to explain Understeer, but here goes:

Your vehicle loses traction on the front wheels, and the wheel just keeps slipping, not letting the vehicle turn.

All wheel drive cars have neither Understeer nor Oversteer, they are just really good

But remember:
the physics of the SP wheels are poo

@ZeroWithSlashedO wheel physics on their own dont really suck, but to get good wheel grip and/or nice properties, they need some setting up.

Simply put, Sideways Traction controls your "grip" when turning. And Forward Traction is the ability to move forward.
Although wheel physics kinda suck rn.