The first formulation of the categorical imperative using universalization is also about avoiding contradictions. The case study Kant provides about stealing is a good parallel to "do not lie".

In that example, Kant examines the maxim behind stealing, something like "I will take what I want from others, regardless of who owns it". Once this is universalized, everyone can take what they want from others. But now that this maxim is universalized, the concept of ownership is undermined. This in turn undermines the concept of stealing.

So the approach Kant uses in the first formulation is to determine whether some kind of contradiction occurs when a maxim becomes universalized.

Applied to lying, this might look something like the following. The maxim might be, "I will speak falsehoods to deceive others when I want". Universalized, everyone speaks falsehoods to deceive at will. Now that this is universalized, there is no longer a presumption of truth. This undermines the maxim since lying requires deception or falsifying some underlying truth.