There are two online interfaces to the dictionary data, instructions on how to use them are included in subsections below.
On the landing page, scroll down and click the green "Sutinku" ("Agree") button, to indicate agreement with conditions of use presented on that page. You can use online translation tools to translate the page before agreeing, if you are so inclined.
On the next page, type the initial letters of the surname you are interested in into the search box. The auto-completed results will be immediately displayed in the drop-down below. You don't need to worry about diacritical marks on Lithuanian letters, for example "C" will match both "C" and "Č", "S" will match both "S" and "Š", etc. You can either select the individual surname of interest from the list, or click the search button next to the box, to display all surnames matching a particular prefix.
Typical results entry contains the surname, followed by an abbreviated list of locations where it was known in the 1930s. After each location there is a number, indicating the number of families with the given surname, recorded at this location, allowing you to quickly see which locations were the most "popular" for the given surname. The number in brackets following the location list is the total number of families, a sum of individual per-location numbers. Most of the entries are followed by a sentence or two decribing the etymology of a particular surname (in many cases they are just guesses). Pointers to surnames of similar origin may be provided as well.
In order to decode the locations, consult the location abbreviation page. To access it, click on "Sutrumpinimai" ("Abbreviations") at the top bar, then select "Gyvenamųjų vietų vardų sutrumpinimai" ("Abbreviations of the location names") from the menu. You probably want to open that page in a separate browser window while using the dictionary, for easy lookup.
Appears to be backed by the same (or, at least, very similar) data, as the dictionary website above.
This site offers two search boxes, "Pavardė" ("Surname") and "Vietovės" ("Locations"). Typing into either of them will produce a list of autocompleted results. Note that this autocompletion does not always work reliably, and may omit matching entries, so it is recommended to cross-reference the data with the surname dictionary. You can also click on one of the letters at the top to see all surnames starting with that particular letter. Also, it is important to note, that differently from the dictionary, you need to use the letters with correct diacritical marks, for the search to work correctly.
You can use "Vietovės" ("Locations") box to restrict the search to only specific locations. Again, auto-completed results will be offered, once you start typing into that box. This is a very handy feature (when it works properly), as you can, for example, use it to display surnames starting with a particular prefix (may be just a single starting letter), known at a particular location. Obviously, only larger locations are mentioned in the database, so if the village of your ancestors is not listed, try the nearby larger villages or towns.
Once you are happy with the entries in the search boxes, click on "Ieškoti" ("Search"), and matching results will be displayed on the panel below them. Next to each of the surnames listed in the results there is an "Išskleisti" ("Expand") link, which can be clicked to display associated locations for that surname, along with the corresponding family count.