what does "line ministries" mean?

Solution 1:

From Revolution at Queen's Park: Essays on Governing Ontario (Ed. Sid Noel, 1997)...

The manner in which the centralization/decentralization issue is structured in terms of cabinet decision-making is closely mirrored in the relationship between the central agencies and the "line" ministries (i.e., ministries such as Health, which deliver programs and services to the public.

I think we can reasonably see the usage as an extension of the more common (in BrE)...

line manager -a person with direct managerial responsibility for a particular employee.

...in that the "line" in corporate management figuratively stretches from boardroom to workers, just as the political/administrative one sits between central government and the general public.

Also note this from Developing Public Policy: A Practical Guide (Bobby Siu, 2013)...

A line department is also known as a line ministry.
The two terms are often used interchangeably; however, in common usage, people tend to use the term department only for an office in the federal government, while ministry is used to refer to offices in a provincial government.

Solution 2:

I have faced the term "line ministries" myself. By checking various contexts, I'd suggest "relevant ministries" as a definition.

Here's an example: National youth development is often the sole responsibility of the government ministry or department where the youth portfolio lies, whereas youth issues should be mainstreamed across various sectors and line ministries such as health, finance, economic development, housing, justice, foreign affairs, education, and agriculture.