"Stripping" presumes there is already a typeset copy that has been photographed, and the film is ready to be stripped; i.e., laid out on a frame for offset printing.

Printers use the term 'stripping in' to mean insertion of typeset material in an existing layout. It is frequently seen on printer's estimates.

It is used where, for example, the film of an existing card, leaflet, etc, is adapted, by cutting and pasting strips of film onto it.

Whether this applies in your case is another matter. If the business card is being produced with 100 different variants, then 'stripping in 100 names supplied' would refer to adapting the layout for 100 different people. However, it's not clear from the context you supply, whether this is the case.

With printing business cards, the process is normally as follows:

Text (like individual names, telephones etc.) is usually restricted to only one of the printed colours. E.g. if it's a full-colour card, three of the colours are identical for all cards (therefore the printing plates remain the same for the whole order), and only the fourth plate (typically the black one) is changed for the different print batches.

To prepare the black plate, one needs to:
1) open the ready file and enter the different names/data (Einfügen) in the preset positions.
"Einfügen der 700 Namen auf der Vorderseite": This is likely to mean that there are 7 batches of business cards with identical design, but with different names (100 cards per printing sheet, where the sheet is probably 100x70 cm, as it holds exactly 100 standard-sized cards). The card is probably double-sided, hence "auf der Vorderseite"

2) Produce the black ink separations/films (through printing to an imagesetter) and use them to produce the black ink plates for each of the 7 batches.

Uhm... I'm a bit confused here. In pre-computer times this was indeed done by stripping (Montage) - a high-precision hand operation. Each card (or block of several cards) was on a separate piece of film, and these were pasted (taped) onto a transparent sheet in a way strictly matching the layout. However, nowadays no one wastes time to strip by hand - you just change the info in the file and print out the ready layout to a full-size, single film sheet. Maybe under "Montage" they include the price of the film (eventually - and plate) for each of the 7 batches? Normally there's nothing else to pay for...

"Montage" may have remained as an item from older times, still used in the accounting of that printing shop.

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References: I'm basically a graphic designer, working mainly for print :)