What is the process teeth go through in the modern world to cause this?
Our mouth has a rich community of microorganisms, among which there are many species of bacteria. Here's a simple overview:
Think about yogurt. Its characteristic sourness is achieved by fermentation of sugars naturally found in milk (lactose, and its components: glucose and galactose) by lactic acid bacteria, such as those in the genus Lactobacillus. These bacteria generate energy by converting certain sugars into lactic acid, as their name suggests. Normally, these bacteria are already present in your mouth, but they don't get to take over the other populations of bacteria, unless you introduce a lot of sugars. The long-term presence of acids in contact with the surface of the tooth may cause demineralization.
What does it mean? Enamel is the tissue found in the most external part of the tooth and is also the hardest substance in the human body; it is mostly made up of minerals (more so than bones, which have many gaps). When the pH (indicator of acidity, the lower, the more acidic) drops after a certain threshold, the acid can start to dissolve the main component of the enamel, hydroxyapatite.
How quickly after consuming a sugary drink for example does this degradation start to occur?
It depends on your oral health. Technically, it can start as soon as sugar is introduced, but if you don't have dental plaque, then there isn't much of an opportunity to keep direct contact of acid to the tooth enamel.
Normally, saliva is able to neutralize (bring back the pH to normal) the plaque in a few hours, and then tooth remineralization can occur, to some extent. On the other hand, if this happens too frequently, then you wouldn't be able to put the minerals back in your teeth.
How does tooth paste help prevent this?
Mainly by removing dental plaque (where bacteria can produce acid in direct contact with the enamel). Fluoride in tooth paste helps both with killing bacteria and, partly, with the remineralization process itself.