Many learning algorithms either learn a single weight per feature, or they use distances between samples. The former is the case for linear models such as logistic regression, which are easy to explain.
Suppose you have a dataset having only a single categorical feature “nationality”, with values “UK”, “French” and “US”. Assume, without loss of generality, that these are encoded as 0, 1 and 2. You then have a weight w for this feature in a linear classifier, which will make some kind of decision based on the constraint w×x + b > 0, or equivalently w×x < b.
The problem now is that the weight w cannot encode a three-way choice. The three possible values of w×x are 0, w and 2×w. Either these three all lead to the same decision (they’re all < b or ≥b) or “UK” and “French” lead to the same decision, or “French” and “US” give the same decision. There’s no possibility for the model to learn that “UK” and “US” should be given the same label, with “French” the odd one out.
By one-hot encoding, you effectively blow up the feature space to three features, which will each get their own weights, so the decision function is now w[UK]x[UK] + w[FR]x[FR] + w[US]x[US] < b, where all the x’s are booleans. In this space, such a linear function can express any sum/disjunction of the possibilities (e.g. “UK or US”, which might be a predictor for someone speaking English).
Similarly, any learner based on standard distance metrics (such as k-nearest neighbors) between samples will get confused without one-hot encoding. With the naive encoding and Euclidean distance, the distance between French and US is 1. The distance between US and UK is 2. But with the one-hot encoding, the pairwise distances between [1, 0, 0], [0, 1, 0] and [0, 0, 1] are all equal to √2.
This is not true for all learning algorithms; decision trees and derived models such as random forests, if deep enough, can handle categorical variables without one-hot encoding.