At the time, the Israelites believed that if someone was having hardships, and they followed the covenant and were otherwise generally good people, that the hardship must be because something that their parents did. There isn't any foundation for this belief, but it was a common belief throughout many cultures around Israel, and it was likely something brought from a neighboring country (or perhaps even from the time that Israel was captive in Egypt). It certainly would hold true that if a father were to choose poorly in life (e.g. if the father was a gambler who ran up huge gambling debts), those debts, from a legal perspective, would be the responsibility of future generations. Typically the verses that people use to support an actually cause/effect relationship of a father's sins causing problems for the children are taken out of context, that is why it is so very important for someone reading scripture to attempt to understand the context, both linguistically and historically before jumping to conclusions about meanings.
Ezekiel is given the parable of three generations of men, grandfather, father and son, whose rewards or punishments are dependent on their own actions. This is consistent with the rest of scripture, however it ran contrary to a common belief, at that time, of how God works.
The language in this story talks about them being put to death. It has less to do with physical life than with spiritual. A sinner who is unrepentant will not have a positive spiritual outcome (everlasting life, Paradise, Heaven, etc.).
God then says that those who repent, who willingly turn toward God, will be forgiven. This is God's Mercy (mercy is sometimes defined as giving someone a second chance, even if they don't deserve it). Similarly, a person, knowing what is right, who willfully turns away from what is right (chooses to turn away form God) will not be forgiven.
This section deals with life choices. Everyone sins, it is how we face that reality that matters. If we choose to reject God's forgiveness (e.g. if we choose to walk away from God as a path for our lives), then yes, we will die (physically and spiritually). If we choose to seek God's forgiveness, then even if we make mistakes, we will live spiritually, forever, even if we die physically.
The people of Israel thought that it wasn't fair that someone who had committed a "big" sin could be forgiven. They also thought that if they did enough good things, that should be enough, however they ignored the intent of the heart (not the physical muscle in our bodies, but the willful decisions made about the core of our existence). God is telling them that because their hearts weren't right, the things they did didn't really matter.
At the end, God is saying that it isn't really about the things they've done, one way or another, it is about their life decisions, such as desiring to follow God that really matter.
Said another way, it isn't about stocking up "brownie points" with God. We don't get points for doing good things, then lose them for doing bad, hoping that we keep a positive balance. Instead, God wants our hearts to be focused on following God, and if we are thus focused, we will do good things generally. If we come to that focus later in life, after doing countless bad things, God is saying that it is never too late for God to forgive. If, on the other hand, we do good things, but decide to turn away and sin, our prior good works aren't enough to save us because our heart decision wasn't in the right direction.