Basically, what it boils down to is imagine a clock traveling at the same rate of expansion as the universe (the Hubble flow). Now further imagine that clock ticks once per year. The phrase "1.5 billions years after the Big Bang" means such a clock will have ticked 1.5 billion times. It's the only way you can do time that makes any kind of sense, when you think about it, since time is literally relative.
Relativity is what makes things weird, because what you're really wondering is if we somehow had a clock on earth that also ticked once per year (standard years like we measure today), how many ticks would it have ticked after the aforementioned clock ticked 1.5 billion times? Such an "earth clock" will have ticked more times, due to the difference in time dilation. If you're still following along, then that means when we say such a clock I first described, the cosmic clock, has ticked 13.8 billion times, this supposed earth-bound clock will have ticked a near infinite amount of time - meaning, from our perspective, the Big Bang singularity is an infinite amount of our time in the past, even though our cosmic clock will have only ticked 13.8 billion times since the event!
Welcome to relativity!
Note this is the same issue with approaching the singularity of a black hole. The clock approaching the singularity will reach the singularity in the time its gravitational acceleration indicates and will proceed normally to the singularity. From outside of the black hole (you have to imagine being able to make this observation) you will never see the object reach the singularity. Such an event is always in that observer's future.
If you think this is bollocks - it's GR's time paradox and is the result of a singularity, which is a strong sign that things have gone awry. BUT - we do know time dilation exists and we know GR works well right up to the point of the actual singularity so while from our perspective the object reaching the singularity probably isn't infinitely in our future, it's very, very far in our future. Likewise, for the Big Bang singularity even though the cosmic clock ticked off 13.8 billion years, from our perspective the event was much further back in time than that.
That's a good question and the Wikipedia has an oversimplified answer here: https://en.wikipedia.org/wiki/Cosmic_time
Basically, what it boils down to is imagine a clock traveling at the same rate of expansion as the universe (the Hubble flow). Now further imagine that clock ticks once per year. The phrase "1.5 billions years after the Big Bang" means such a clock will have ticked 1.5 billion times. It's the only way you can do time that makes any kind of sense, when you think about it, since time is literally relative.
Relativity is what makes things weird, because what you're really wondering is if we somehow had a clock on earth that also ticked once per year (standard years like we measure today), how many ticks would it have ticked after the aforementioned clock ticked 1.5 billion times? Such an "earth clock" will have ticked more times, due to the difference in time dilation. If you're still following along, then that means when we say such a clock I first described, the cosmic clock, has ticked 13.8 billion times, this supposed earth-bound clock will have ticked a near infinite amount of time - meaning, from our perspective, the Big Bang singularity is an infinite amount of our time in the past, even though our cosmic clock will have only ticked 13.8 billion times since the event!
Welcome to relativity!
Note this is the same issue with approaching the singularity of a black hole. The clock approaching the singularity will reach the singularity in the time its gravitational acceleration indicates and will proceed normally to the singularity. From outside of the black hole (you have to imagine being able to make this observation) you will never see the object reach the singularity. Such an event is always in that observer's future.
If you think this is bollocks - it's GR's time paradox and is the result of a singularity, which is a strong sign that things have gone awry. BUT - we do know time dilation exists and we know GR works well right up to the point of the actual singularity so while from our perspective the object reaching the singularity probably isn't infinitely in our future, it's very, very far in our future. Likewise, for the Big Bang singularity even though the cosmic clock ticked off 13.8 billion years, from our perspective the event was much further back in time than that.
Weird stuff!