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enter image description here

Left picture is showing that the vector data can not be separated by a linear line. but after You are using some basis function $\phi (x)$. it transforms to the right picture.I kind of understand the basic concept but not exactly understand how(mathematically) it transformed to the right picture.

anyone knows about this topic? any other resource will be good. I'm not sure how to search about this topic.

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First your title is not accurate, we are not transforming model, but data.

For the topic, you can search basis expansion as key word. In the example shown, search kernel PCA is another option. Note, wikipeida on kernel PCA has very similar figure.

Some related resources in CV (with different basis, most polynomial basis)

Best basis set for polynomial expansion

What's wrong to fit periodic data with polynomials?

How do I go about increasing model complexity?

Why are there large coefficents for higher-order polynomial

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