I'm not sure I follow your question, what are you referring to with "solve this?"
I usually find DFS/BFS applicable when I'm dealing with data that forms relations like "children" or "neighbors" or "connected". Sometimes that's explicit because the data is already in a tree or graph structure, sometimes it's not. And often rather than looking for a particular element, I want to iterate over all the data connected by the relationship to do something with after as a list/set/map of the data or some of its properties.
A concrete example from a side project (it's more fun than Real Work examples, and people have given other examples in the thread anyway), I was writing a client for the board game Go. If you're not familiar, the data is just a 2D grid (easy to interpret as a graph), black/white stones get placed on coordinates like (3,4) marking intersecting lines. A single stone forms a connected group with another stone if both are the same color and separated horizontally or vertically by one. Groups have a count of "liberties" representing the number of unoccupied points the group can expand to if a stone of its color is played there -- a single stone by itself in the middle of the board will thus have 4 liberties, if you connect a stone then that group now has 6. If you cause an opponent group's liberties to go to 0 by surrounding it, the whole group dies, and its stones are removed from the board with those points becoming unoccupied again. Suicides (causing your own group to hit 0) are ok only if they capture the opponent first, otherwise are invalid moves.
Handling the game rules is easy with DFS/BFS. All you need to start with is a function like get_neighbors(position) which on a grid is trivial, being the 2-4 surrounding points depending on whether position is an edge/corner. Then you can use DFS/BFS (doesn't matter) and several lines of code later you've got a function like get_group_member_positions(start_position) that gives you every member stone's position no matter which one you query first. Now you can make a function count_liberties(group_positions) -- which can be solved as another depth-first traversal problem over the neighbors of each member of the group that are empty and haven't been counted already.
Even less code this time though; have you ever had to write 4 lines of code like "for el in list, for child in el.children(), if child blah, do something"? That's a DFS, just not a general one since you know how many layers there are (or that you care about), and it assumes children aren't shared between els.
From all that you can then have a function like get_stones_captured_by_move(move) that tells you what (if any) stones need to be removed if move were to go through: for each neighbor of the move, if any are the opponent's color, count the liberties for that neighbor's group and if it's 1 then those will die when the move is played (takes the last liberty). If it's all one big enemy group around the move you'll have to account for duplicate positions to remove, but those are minor optimization details. Similarly the fact that everything is recalculated all the time, that could be optimized with more storage.
BFS/DFS are just basic building blocks here.