Why is it important to specify your source of funding in published papers and posters?
Several reasons.
Conflicts of Interest. If you're doing climate change research that's funded by The Society of People Who Think Global Warming is Poppycock, you need to mention this. Organizations with a clear interest in a specific outcome are much more likely to produce research that affirms their desired outcome than they do those that refute it, regardless of the actual truth of the matter. Researchers may perform ethically compromised science to satisfy the funding agency. Since organizational bias (and pressures they exert on their researchers) may not be known until after the fact, this information should always be mentioned so that we can always make the best judgment of the results in light of current knowledge.
Part of the point of funding you is so that the organization's name is out there and associated with (hopefully good) research. If McDonald's funds your race car, you're expected to put a sticker or something on it with their name on it. Grants often include specific language requiring you to mention the grant in any work that directly utilizes it.
So the grant agency can track work done with their money. These agencies want to know that their money is being used appropriately and for good projects. They don't want to just throw large sums of cash out there willy-nilly. Money isn't meaningless; it's worth something. This may even be required by law: if they are funded by a government, the government is going to want to know how the money is being used. Without such tracking, funds can be misappropriated or embezzled, or wantonly wasted on clearly sub-standard proposals.
There is also a technical reason: to prevent a certain form of a fraud.
Imagine you have grants from agencies A, B, C, and D. For all of these you will need to submit final reports that detail how you used the funding and what you achieved with it.
If the papers didn't contain any information about the source of funding, a dishonest academic could just take the list of all papers produced in their group, and submit a report to agency A claiming all of them were solely funded by the grant from A, and a similar report to agency B, etc. Each agency would think that you have produced a good number of papers in comparison with the volume of funding they are aware of, and they would be happy.
To prevent this, the agencies require that you can only list papers that explicitly acknowledge the source of funding. This way the above type of fraud is not possible. In essence, to submit convincing final reports you will have to have a research output that is somewhat proportional to the total amount of funding, even if you have multiple independent sources of funding.
They might be completely meaningless to you, but they may be meaningful to someone, and if you don't have it as a standard policy, you won't catch those edge cases.
In my experience, there are two major reasons, both of which are important:
"Credit Tracking" - What grant funded what is essential both for claiming projects on grant reports (many funding agencies won't let you claim a piece of work comes from a particular project unless it's in the funding line) and for the public perception of the program. To use two examples from my own work:
This project was supported in part by a research grant from the Investigator-Initiated Studies Program of Merck Sharp & Dohme.
In this particular case, Merck is actively proud of their investigator-initiated program, considers research publications to be the primary output, and wants people to know that.
This work was funded by NIH MIDAS Grant XXXXXXXX...
In this case, the NIH has a specific funding mechanism set up for a specific purpose involving a network of researchers. This is, essentially, marketing. "Remember that program we set up? This is part of that."
"Disclosure" - Knowing how a paper was funded can provide useful context. There was once a paper that made very little sense in terms of what they were doing until I realized that they were funded by a particular company, and essentially talking about that company's product without saying it's name. Then, suddenly, the specificity of what they were looking at made perfect sense.
It's also useful context - a study of the efficacy of an alcohol-based hand rub will be read in a different light if its funded by the NIH, a soap manufacturer, or the maker of the hand rub. Even if it's good science it's useful information, and disclosing it for everyone and ignoring it most of the time is better practice than hoping sometimes if it's relevant we find out about it.