Why does a loving God send people to hell?

God is infinite in all of His attributes.  God is infinitely good, for example.  God is infinitely powerful.  God is infinitely present.

God is infinitely Just.
God is infinitely merciful.

There's the contradiction.  God cannot be both infinitely merciful, in that He has mercy on everyone, and also infinitely just, in that all wrongdoing is punished.

Justice and Mercy are, in some sense, opposites.

So how does God, who is both infinitely Just and also infinitely Merciful, maintain both of those qualities?

By punishing all sin, and by showing mercy to everyone.

Thus, the punishment for all sin was meeted out onto Jesus, who is God Himself.  In this way, Justice is served.  But mercy is also extended to all persons because now that justice is served, there is no need for further punishment.  That's infinite justice and infinite mercy.

Now, you may ask, why does He send those who do not believe in Him to hell?

Glad you asked.  He doesn't.  God loves everyone with an infinite love.  He allows us to chose our paths in life. If we chose to reject Him, He honors that choice.  If we spend our whole lives, therefore, rejecting God, God will honor that choice.  He will not force us to spend eternity in His presence.  That's how much He loves us.

But God is the source of all goodness in all of existence.  To be away from God is to be away from everything that is good.  God doesn't send us to hell...  God has gone to amazing lengths to save us from hell... even taking the punishment of our own sins onto Himself... but if we chose hell over a loving God, God honors our choice.