Question

Why is it that Alpha particle radiation is not dangerous outside the body, yet incredibly dangerous when ingested?

Answer 1

When the alpha radiation source is outside the body you have your skin protecting you, the alpha particles are easily stopped and, since the skin is basically dead cells, it doesn't harm you (much). Although it is relatively harmless outside, when the source is inside your body, the particles have an increased possibility of striking essential molecules (DNA, proteins), possibly altering them and, sometimes, making some atoms in your body reactive.

Answer 2

Alpha molecules can be stopped by paper, your keratinised, dead skin does an easy job of this although you are still at risk of developing skin cancer through mutation. Anyhows. When ingested it passes the mouth, esophagus, stomach, intestine, anus and out again. all this time it is in contact with living cells of our body and those cells are under direct fire from the alpha particles. If these particles effect the DNA of the cell it effects the protein synthesis of the cell with is v.v.v.v specific which directly effects the cells ability to do its specific function in the gut etc. We can also have rapid and malignant growth of the cell which passes on this harmful and useless change in protein synthesis but also developed tumors and cancers which are generally not benign and can lead to serious illness and death if not quickly treated.
The main reason overall is outside we have our dead layer of protective waxy skin, inside we have nothing of the sort, if we did we would have a very hard time absorbing our nutrients to survive.

Hope that helps