Why does The Institute kidnap people?

What do they hope to accomplish?

The Institute wants to rebuild civilization. However, they're insular, and largely cut off from the outside world. They want new civilization on their terms. The shape of this future civilization is largely colored by how the Institute's members live today. They are a scientifically literate, technologically advanced, strongly hierarchical society, with humans performing "thinking" jobs, and synth slaves performing manual labor and any tasks "beneath" humans. Human outsiders are not trusted, and generally thought of as violent barbarians. Killing them is perhaps on par with killing wild animals. They can be used to advance the Institute's goals with few moral qualms.

Most of their goals revolve around building a robust new society (under their terms, as outlined above).

• Remain hidden. Infiltrate & spy on outside organizations. Destroy major enemies (BoS, Railroad).
• Develop crops that perform better in post-war conditions.
• Improve humanity, animals, and living conditions through technology and bio-engineering.

Of course, like any large organization, there are disputes on what goals should be persued, and how. However, thanks to the strict hierarchy, the directors and Father have the ultimate say in the direction of the Institute. For example, one of the advanced systems terminals contains entries on research proposals. A proposal on Human Cybernetics is listed as rejected by the Director, despite the fact that the Institute pursued these avenues in the past.

Why do they kidnap and replace people with synths?

Because it advances their objectives. In many cases, this means spying on outside groups or infiltrating organizations they perceive to be a potential threat. In others, they're running experiments.

For example, one terminal outlines the Warwick Homestead Initiative:

Mission Statement The Warwick Homestead Initiative was conceived to facilitate field testing of various genetically-modified specimens in the unique climate of the Commonwealth. It is hoped that we gain insight into the effects of trace radiation on the growth and developments of said specimens.

We of the BioScience division are in full agreement on the following hypothesis: subjected to the proper levels of ambient radiation and soil ph such as those found in the exceptionally fertile soil at Warwick Homestead, our modified seed specimens are likely to exhibit accelerated growth rate and a two-fold increase in size.

Project Implementation Stage 1: Using genetic manipulation, we will develop a unique breed of Cucurbitaceae with similar characteristics to those commonly farmed in the Commonwealth.

Stage 2: Acquire Roger Warwick, patriarch of Warwick Homestead, and conduct a series of intensive interrogation sessions to learn all we can about his life and family.

Stage 3: With the intelligence gathered in Stage 2, create a synth replica of Roger Warwick and embed the unit on-site at Warwick Homestead to oversee the operation directly. SRB will handle logistics of this aspect of the initiative.

Stage 4: Begin covert deliveries of prototype seed batches for planting.

Stage 5: Collect observational data from embedded unit.

Stage 6: When sufficient data has been collected, retrieve synth unit and specimens for lab study. Purge all surface evidence of the initiative.

This plan involves kidnapping and replacing with synths, as well as eventually killing a number of innocents. But it's not done just for laughs, it's strictly to advance their goals.

What do they do with the missing person?

I don't know if it's ever explicitly spelled out, but with their nonchalance toward killing outsiders, it seems likely the replaced person is "disposed of" as soon as they're no longer useful.

Source 1 - Fallout Wikia

Source 2 - Gamepedia - Institite terminals

Just wanted to add that either a holotape or terminal in the Bioscience division specifies that they experiment with the FEV on kidnapped people.

In my opinion, the answer to all three questions is "a giant plot hole." The other answers here are an attempt to guess based on incomplete information.

The writers of the game attempt to try a New Vegas plot with 4 factions grasping for control.

Ceasar's legion (The Institute) is a ruthless organization intended for players who want to be the bad guy. They steal destroy, and plot with no thought other than their own expansion. They intend to burn the land and replace everything with people they bring along. An unlikely leader hangs on through force (corsairs), but the slave revolt is on the horizon. Somewhere along the line both the Legion and the Institute got nerfed to make them more family-friendly. Little hints are all that remain of the original message.

New California Republic, NCR, (Brotherhood of Steel, BoS), is an outside force for those who want to play justice police. In both games they determine the locals can't govern themselves. This faction is for players who want no compromises. You must be judge, jury, and executioner because your cause is just.

New Vegas computer guy (the railroad) are a limited organization with big plans. Both start out small but have a plan and a secret weapon to seize Vegas. Intended for your sneaky player who likes the idea of controlling things from the shadows.

Lastly, the player faction. Here the player cheats another faction claiming it as his own. In Vegas you take the computer's robot army, in Skyrim (sorry, Massachusetts) you build the minute men to become the most powerful force, then do nothing with them. It's intended for the "good" player with a live-and-let-live attitude.

I wish I had better news for you but I don't. The plot and in many cases the stories here are incomplete. Chris Avellone wrote the great Fallout 2 along with the well plotted New Vegas. I can't find anyone who claims credit for writing Fallout 4 (45 seconds of Google). I know I wouldn't have posted my name if I stole the factions and the plot (something important is taken, I must join a faction to recover it, it turns out not to be important, I lead my faction to dominance over everything). Especially if I did as bad as job as this.

See also http://www.forbes.com/sites/insertcoin/2016/01/08/why-ive-soured-on-fallout-4s-ending/