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.