Nobody really knows how type Ia supernovae detonate (or deflagrate) - there are a number of possibilities. The "vanilla" possibility is not what you state in your question, it is that the white dwarf accretes sufficient mass that it approaches the Chandrasekhar limit and becomes dense enough in its core to commence carbon burning.
However, the emerging diversity that is seen among type Ia supernovae, once thought to be a single population, suggests there may be other possibilities. There is some evidence that white dwarfs may explode at masses well below the Chandrasekhar limit. If a white dwarf in a binary accretes enough He-rich matter, this can become compressed enough to ignite He burning near the surface (this happens at a lower density threshold than Carbon burning). This then drives a shock wave into the white dwarf and the compression caused by this can ignite the carbon.
Why does the He "explode"? Well, the accreted He will form an electron-degenerate layer at the surface. A fundamental property of this degenerate gas is that the pressure is independent of the temperature. Thus, if the He ignites then at least initially, the temperature goes up but the pressure does not. Since the He fusion rate depends on something like $T^{40}$ this allows a runaway reaction to develop that could be characterised as an "explosion" in the surface layers.
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