First of all, let's not hold any illusions that this practice isn't problematic. While the Jedi never forced it on anyone, it was indeed institutionalised by the Galactic Republic, and on hindsight to historians it might have indeed been a controversial issue in the later years when the Order grew increasingly arrogant and dogmatic. The Galactic Empire did use this practice as an excuse to spreading hate on the Jedi and raise support for their extermination. Its original intent, however, was nonetheless noble and pure, and the galactic commonfolk are used to being supportive of it.
A little history and analysis may help lead you to the likely reasons for the lower than expected resistance to this practice (from the perspective of an Earthling). The Old Republic was engulfed in a war between Jedi and Sith when a branch of ancient Force practitioners discovered the dark side, eventually leading to the Old Republic's collapse.
The galaxy was liberated from that ancient war thanks to the Jedi, leading to the founding of the Galactic Republic. As the galaxy disarms itself after the war, the Jedi would also take the place of the military as the galaxy's peacekeepers. Being saviours of the galaxy, founders of the Republic, keepers of the peace and defenders of justice do wonders to your public appeal. When the novelty of being the saviours and founders wear away with time, it is replaced by the aura of being an ancient order, and a sense of rightness to sending your kids to them because "they've always been there, and we've been doing this since the time of our ancestors and founders".
But only Force sensitives can be a Jedi, and to prevent a repeat of history, prevailing Jedi beliefs prefer inducting into the Order Force sensitives who have not matured enough in life to form attachments (which eventually leads to the dark side - that's why they initially refused to train Anakin because he is too old). Since they are the Republic's main and most effective peacekeeping force, and this is the recommended way to maintaining their numbers, the Republic Senate would naturally want to support such a policy. This is probably what led to the institutionalisation of this practice of taking them young. Basic medical examinations within the Republic's borders would check midichlorian counts - anomalies are brought to the Jedi's attention, who will record these entries and visit the babies. The babies are first raised by their parents for a time, until early childhood when they are brought into the order to be formally trained. In Rebels, Kanan once mentioned not having seen his family for a long time - indicating he was old enough to remember them (although Force sensitives in general tend to mature faster mentally).
So that's probably how this practice started. But why did people accept it? One out of three reasons, depending who you are:
- You're poor. Sending your child to the Jedi reduces your own living expenses and gives your child a much better life than what you can ever give. The government may even give you good compensation for life too. Imagine if you're a peasant in North Korea and the government offers to support your family in exchange for taking your child because he has talent to be a good IT guy. Will you accept it? (This is a true story, North Korea pays well to keep their Internet running).
- You're rich, powerful and/or influential. The Jedi is the best thing in the galaxy short of being Supreme Chancellor. It raises your status!
- You come from a more primitive or more religious society. In such societies, Force-sensitives (before they knew what is the Force) tend to become spiritual leaders like shamans, witch doctors etc. Today, becoming a Jedi is just an upgrade of that. You're even more revered in your native society.
Basically, lots of people have good incentive to send their eligible kids to the Jedi. The only ones who don't are people who hate the Jedi, and that generally are enemies of the Republic. No matter how bad the Jedi Order is in its final years, this practice is still less infringing on sentient rights than, say, the First Order's stormtrooper program.
Note, the above is mainly my own analysis and not evidence-based. But nonetheless, I do think this is a satisfactory rationalisation of the reasons that lead to this practice being existent in a galactic civilisation where slavery is banned.