Here's why -
Let's start with the Declaration of Independence, which is available to read HERE.
The second sentence of the Declaration reads,
We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness.
It mentions three unalienable rights:
the pursuit of Happiness
I will only discuss the first one here. It is relatively easy to define, and the other two can have potentially unlimited definitions.
Life (noun): the condition that distinguishes organisms from inorganic objects and dead organisms, being manifested by growth through metabolism, reproduction, and the power of adaptation to environment through changes originating internally.1
Life in this discussion then applies to all human beings (men, mankind, humankind), wherever situated, at whatever age, from conception until natural death. The pro-life and pro-gun positions generally adhere to the following definitions:
Pro-life (adj): the belief that all human life is created equal regardless of size, level of development, education, and degree of dependency.2
Pro-gun (adj): favoring the right to own guns and opposing legislation restricting this right.3
Why are the pro-life and pro-gun positions compatible?
The pro-gun position embodies the concept that innocent human life is worth defending from harm, whether that harm comes from a common criminal, a mass murderer, foreign invaders, or rogue government. It is not really about guns, per se. It is just that the gun is the best tool to defend innocent life.
The pro-life position, analogously, embodies the concept that innocent human life is worth defending, even in early developmental stages. Scientific terms such as zygote 4 and fetus 5 can be used to cleverly dehumanize these early stages of human life. But they are still fully distinct innocent human beings.
So, pro-life and pro-gun are really two sides of the same coin, protection of innocent life. One, before birth; the other, at any time from birth forward.