I'm currently studying Boosting techniques in Machine Learning and I happened to understand that in Algorithms like Adaboost, each of the training samples is given a weight depending on whether it was misclassified or not by the previous model in sequential boosting.
Although I intuitively understand that by weighting examples, we are letting the model pay more attention to examples that were previously misclassified, I do not understand "how" the weights are taken into account by a machine learning algorithm. Are the weights taken as a separate feature? I want to know "how" a simple machine learning model, lets say a decision tree, understands that a particular example carries more weight and therefore needs to be given more attention/ classified correctly.
Can someone please explain to me in simple words/equations how weighted training samples are handled by machine learning algorithms? How do they force themselves to classify samples with higher weights correctly? Should there be modifications made to existing ML algos to account for weights?