Why Wiesler begins to act like that?

Early in the film, Wiesler found out that Minister of Culture had romantic feelings for Dreyman's (the screen writer) girlfriend. He wanted something to use on Dreyman so he could "eliminate" Dreyman as a romantic rival.

Wiesler's boss/friend liked the idea of spying on him. Because he looked at it as a way to get a promotion.

So Wiesler started to spy on the couple. But he found himself liking the couple and he learned that Dreyman wasn't a subversive to the government. All Dreyman wanted to do is to let West Germany know about the suicides in East Germany. Because the government controlled press didn't report suicides.

Wiesler knew what was going on. As a Stasi agent. He reported as little as possible. And when the Stasi were going to arrest Dreyman for the news story and having an unregistered typewriter. So Wiesler took the type writer to save Dreyman from being arrested.

And that's when Wiesler's boss/friend send him to the dead end job opening up mail. Because his boss wouldn't be getting that promotion.

Weisler learns that his bosses want him to spy on Dreyman not because he is a threat to the GDR, but for petty personal reasons -- because a higher-up wants Dreyman out of the way so he can get Dreyman's girlfriend.

Before this, Weisler had been blindly loyal to the GDR and the Stasi's activities. But the realization that the men in power are using their power for their own ends, not for the GDR, makes him lose his idealism toward the GDR.

The scene in the cafeteria extends this theme. A worker faces severe punishment for telling a Honecker joke. Another example of the Stasi higher-ups destroying lives (or at least thinking about it) just because they have the power to do so, not to "protect" the GDR.

Once Weisler starts questioning why he does what he does, he is open to having empathy to the couple based on what he hears when he spies on them.

You must be the change you seek in the world. -- Gandhi

Nicely put. I worked there in that region then for the UN, and would suggest also two other reasons.

(1) They were not just any ol' couple but a high-achieving artistic couple and Wiesler had seen and come to appreciate their art, and he liked the kind way that they chatted and lived. In most of the East European countries the artistic and academic communities kept their heads down and tried to remain refined. When the wall came down, all across Europe, these artists and academics were the ones that came to power - most prominently in Prague of course with Havel the "poet king".

(2) Through the 80s it was growing obvious in all those countries that the Soviets had built an economy that no longer worked. Gorbachev met with Reagan in Iceland in 1986 as a direct result. Aparatchiks all across Europe throughout the period of the film were making their choice. Wiesler was far from alone.

When East Germany DID open up there was considerable shock in the west at how backward it really was.

Wiesler was a perfectionist who was loyal to the government. When he started spying on the couple, not only he realized that there wasn't anything condemnable on Dreyman (he was just a victim of Minister Hempf's personal interests) but also he became fascinated by them and their lives. He helped them because deep down he was a good, decent human being and not just a ruthless Stasi agent.

This is - excuse me - a damn fine cup of coffee.

There was a scene quite early on when Christa was sexually assaulted by the minister in his car. When they arrived at the writer's house, Wiesler set off the doorbell to get him to discover Christa's liaison. His intention was to set off a chain of events so that Dreyman would do something and expose his "anti-state" feelings. This was his expectation: all men are like that. Instead, when Christa asked him to hold her, Dreyman did. There was real love, real emotions that he can see in the couple. Wiesler had to get off with a prostitute and that contrasted the falseness of his life and the rich life that was happening downstairs. That was the beginning of his changed attitude towards the couple.

I thought the transformation took hold in the elevator scene with the little boy. Here was this cute little 4 year old expressing curiosity about the Stasi and repeating a negative remark his father made about state security. Wiesler’s first instinct was to question the boy for his father’s identity. The instinct was so automatic, you felt like he had probably done this before: using an innocent child as an unwitting informer against his own family. Wiesler stopped himself from proceeding on this course and I felt this was the the point he started to turn

Very subliminal. Very artsy.
A low budget film has it's limits. Plus it's a fictional story. Contrary no Socialist (SED nor Stasi) real life case is known. You know afterwards ppl wanna be the celebrated media heroes. Yet silence. Look at today's Left-Wing Brood, it's the worst doctrine. PPL running ahead for that ideology have no righteous life, no sense for common normalcy nor justice. Very own gain is all. Even Wiesner is on the low-key pissed his old mate he backpacked throu study grades got promoted ahead of him.
Weasel Wiesner should have died instead the poor Actress being worst off of all.

I found it hard to believe as well with the introduction of the movie and how he wrecked innocent ppl lives for approx. 20 years!
Encounter with a young boy in an elevator where multiple Stasi Agents lived as the prostitute revealed. And citizens around are so aware openly talking about them that even kids repeat. Using kids to expose parents was a well known basic tactic he also had to teach new recruits. And since parents knew they were very cautious with their kids. This scene would have never happened in 80ies, maybe 60ies!

The art talk wasn't all that. I was expecting way different for such a dramatic change of mind-set.
He got insight on true love in mid mid-/late 40ies. Yes but I'm sure they had not just preferred access on prostitutes but first and foremost fitting women for marriage. After all no woman seeks a career in prostitution but wanting to have a good status man and family. Portraying these dudes as single meaning woman opposite also staying single. Not accurate in 80ies, esp socialist countries! This society trend just started earliest in 90ies with ppl having way to high expectations on media stars. And then Internet searching forever for Miss Perfect.