How does Jesus show love by suffering? I wonder, how does the suffering of Jesus/God show love?

I do not believe that Jesus suffered to forgive our sins. If God is love, God doesn't need to make a terrible sacrifice to forgive our sins. God can forgive without it. This explanation is really inconceivable to me. 

It seems to me like part of the answer is to show that God suffers with us and that we don't suffer alone, but this answer doesn't completely satisfy me either.
Short answer:

The reason is, that Jesus' suffered and died in our place. 

This is how the bible puts it in Romans 5:6-8 ESV: 


  6 For while we were still weak, at the right time Christ died for the ungodly. 7 For one will scarcely die for a righteous person—though perhaps for a good person one would dare even to die— 8 but God shows his love for us in that while we were still sinners, Christ died for us. (emphasis added)


By way of further explanation:

The bible tells us that suffering and death entered the world (and persists in it) because of mankind's sin - that is, our deliberate rebellion against God's perfect way of doing what is right. Sin is massively problematic, not just in it's short term effects (immediate victims), but in it's long term consequences (for instance, our very identity becomes limited by our warped behaviour) and also the fact that, by it's nature, it excludes from (our) consideration the Great Source of Life, Truth and Healing who could remedy the situation - the loving Heavenly Father and Creator that we have rebelled against. Suffering (and ultimately death) is both a consequence of sin, but also a judgment against it - God uses it to discipline, restrain and ultimately stop us from multiplying sin and wickedness indefinitely.  If God were to 'just forgive us', it would not solve the underlying problem of the sin nature in us - we would continue to sin more and more and continue to suffer the consequences for doing so. The fundamental problems to solve, are our separation from God and our sinful nature.

Though we would not turn to God to solve our sin problem, God did not leave us to our devices without a remedy, but came in the person of His Son, to live amongst us - showing us what the Heavenly Father is like and also experiencing our humanity - including weakness, suffering and death.  The suffering and death that Jesus endured were not a consequence of His own sin, as He was perfect in all His ways and always did the things that please the Heavenly Father, rather, they were entered into willingly as a substitionary offering to take on the suffering and death that all mankind deserves.

As Isaiah puts it:


  4 Surely he has borne our griefs and carried our sorrows;
  yet we esteemed him stricken,
      smitten by God, and afflicted.
  5 But he was pierced for our transgressions;
      he was crushed for our iniquities;
  upon him was the chastisement that brought us peace,
      and with his wounds we are healed.
  6 All we like sheep have gone astray;
      we have turned—every one—to his own way;
  and the Lord has laid on him
      the iniquity of us all. - Isaiah 53:4-6 ESV


Since Jesus has paid this price, we now have the opportunity to renew a relationship with God and receive 'salvation' - this word contains many ideas, but it's essence is directed at the complete reversal of the problem of sin and the consequences of sin including suffering and death.  We can by faith, receive healing from God and eternal life.  Because of the long-term effects of sin on this world, we may still experience temporal suffering to a degree (though it's impact will be greatly lessened to the faith-filled salvation-receivers) and physical death, but we have a promise of resurrection to a new life completely free of suffering and death - all because of what Jesus has done.

An Illustration: 

You may still be thinking, how does this 'show love'? perhaps the following will help -

Imagine being someone trying to pay a mortgage off on a house.  And through your own fault - even against the advice of those who love you, you spend way beyond your means to the point that you run up large amounts of other debts and can no longer afford to buy even food.  You will suffer in the short term as you haven't kept enough money for food.  You may experience further suffering as various items are repossessed that you cannot make payments for.  You will also suffer anxiety and apprehension that you will be evicted if you cannot make your mortgage payments.  Now imagine, that the one who loves you the most, sees your distress and pays all your debts including your mortgage - the cost (suffering) to them is large, they had no obligation to do this, they did it anyway. They further offer that if you sign over the deeds of your house, you can come and live with them permanently in a much better place that is entirely free of the consequences your mistakes.  There is a wing of their mansion already prepared for you, all you need to do is sign over the paperwork and do a bit of spring-cleaning before you move - the one who loves you will even come and help you do the cleaning! Of course, you are not forced to this, you could just re-mortgage your house to finance your previous lifestyle, and spurn their kind offer; You could conceivably even refuse their help in the first place and insist that you will (somehow!) pay your own debts. In both these latter cases, the end result will no doubt involve a great deal more suffering.

That is a very limited and inadequate picture of what Jesus has done in our place.

Disclaimer: though I am a Protestant, I seek to write not (merely) from that perspective, but from an orthodox (in the wider sense) Christian perspective that includes the Catholic view. (as well as Eastern Orthodox etc.)  If I have not faithfully done that, I welcome any correction.