A lobbyist's task is to convince the politician that a certain position would be advantageous to the politician.
A lobbyist is normally representing a group of people. So part of the deal is "if you adopt this policy, the people who I represent will support you, both at the ballot box and through campaign contributions."
Secondly the lobbyist can argue "If you adopt this policy, it will benefit a wider group of people, and it will be electorally advantageous to you.
For example, a mining consortium may lobby for exploration rights in a certain region. The lobbyist says (in blunt language), "If you use your influence to grant exploration rights, the companies that I represent will be minded to contribute more to your campaign funding. Moreover, mining will create jobs and bring wealth to the region and this will make you more popular among the people who will become more wealthy."
At the same time an environmental NGO might be lobbying "If you use your influence to block exploration rights, the organisations that I represent will be minded to contribute to your campaigns (perhaps not financially but our NGO has influence in the community and we will speak positively about your actions) And moreover, by preserving the beauty of nature you will become more popular among people who value wild spaces"
Essentially, the lobbyist is being an advocate, arguing a case, and trying to persuade the politician that a particular position is the right thing politically.