For many reasons.

First of all, 1.5°C spread out over the entire atmosphere means that there is significantly more energy inside the atmosphere. This manifests as extreme weather such as storms, floods and hurricanes.

Second, this increase in temperature increases the amount of polar ice that melts, which increases the sea levels, which can lead to more flooding or the disappearance of entire regions or island nations.

Third, some ecosystems are very very sensitive to temperature changes, because they have evolved under a certain temperature for a long time.

Global temperature - by definition - includes the entire globe.

The amount of energy it would take to increase the entire globe by 1.5°C is absolutely staggering.

Adding a mind-bogglingly large amount of energy to a system makes the system go crazy.

Because it has a knock-on effect. A 1.5⁰C increase literally by itself might actually be nice in some places.

However, an increase in global temperatures can completely mess up the weather (hence the increase in forest fires, and hurricanes this year).

As I know we live now in time of the one of the most mass extinction of live creatures. Mainly insect are dying, which can cause that plants will die and then we all die. And there are many insects which very sensitive to temperature.

Source: https://www.sciencedirect.com/science/article/abs/pii/S0006320718313636

1.5°C is the average across the entire globe, but because of the way global warming influences the climate, this will result in much more extreme temperature variations on a local level.

For a somewhat hyperbolic example, imagine that wherever you live ranges between 60-80°F on an average year, the average temperature then would be ~70°F. If the temperature range were to change to 22-120°F due to climate change, then the average temperature would only be one degree higher but temperatures on any given day could be unbearably hot or cold.

Even for a human body 1.5° is a lot of change. A normal human body temperature is 36.4° to 37.2° and having a fever would put you around 38° or higher. Your normal days of life are suddenly all 100% sick days.

You may be thinking that 1.5 degrees isn't that much. Because sure, whether it's 30° or 31.5° on any given day doesn't make much of any difference and you probably wouldn't even notice.

However, it's about global averages. The average temperature of Earth as a planet is the average of all temperatures measured everywhere on Earth near ground level on 365 days a year. If that average changes, it's because a large number of individual data points have changed significantly. While in some places and on some days it may have been colder than the year before, in most places and on most days, it was warmer. Not a lot warmer, the differences year-to-year are small. But decade by decade, the changes should average out - which they don't anymore.

There are a lot of heat sinks on Earth, most significantly the oceans. They absorb huge amounts of energy which is used by the lifeforms living in the oceans (plankton). This ecosystem is attuned to a certain amount of energy coming in from the Sun. It has evolved to use that energy in the most efficient way and thus to keep the temperature of the water steady. It takes vast amounts of energy to heat them up significantly because they are of course massive bodies of water. In the past, when natural causes changed the amount energy, it was usually at a rate slow enough for the ecosystem to keep up and adapt. So if you measure the average temperatures of the oceans (or rather temperatures just above sea level) and you see a rise, something is out of the ordinary. It means that there is more energy going into the oceans than they can absorb and the change is happening too fast for the ecosystem (and evolution!) to keep up with.

I'm no climate scientist so I can't tell you why the numbers are these exact numbers (like why is the absolute upper limit 3 degrees and not 4). But there are by now some quite sophisticated climate change models based on decades of climate research and huge amounts of data about past changes (going back not only decades but millennia). These models can simulate different outcomes for different future scenarios. They're not infallible of course and they are man-made. But these models are used to predict what would happen if temperatures increase by different amounts and that's where the upper limits come from.

Because the weather swings back and forth, hot and cold. When you put more energy into it, these swings get bigger but also trend towards hotter. This means that you get hotter summers and occasionally colder/wetter winters, but each summer will be the hottest summer of your life from now until you die. Worse: a lot of our infrastructure relies on weather patterns and ocean currents, and those are greatly influenced by the evaporation of water in hotter locations. With greater heat, you melt more glaciers, ice caps, and ice sheets. This additional water will break a lot of these patterns, and will eventually make some parts of the world (like Europe) that are very amenable to human living into places that can no longer support agriculture.

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