There are several key elements to help increase newsletter signups on a landing page ... and there's kind of an order to their level of importance ... so, I'll try and present them in that order.
Before I answer this question, NOTE ... where your traffic is coming from and that quality of that traffic is the most important element to increasing signups. If you send garbage traffic, it'll never convert.
Assuming your traffic source is good, the first thing I'd look at is the Lead Magnet. If you're unfamiliar with that term - a Lead Magnet is an incentive for someone to give you their contact information ... it "pulls the lead in." For example, discounts, checklists, whitepapers, videos, ebooks, etc are common Lead Magnets.
Having said that - if no one wants your Lead Magnet, it wont matter how good the rest of your landing page is ... it'll never convert well.
Here's a hint: Many times, just changing the title and/or the picture (aka "hero shot") of your Lead Magnet without actually changing it's content can help increase conversion rates dramatically.
The second thing I'd look at is the Headline and, if you have one, the Sub-Headline. Maybe it needs to be shorter. Maybe it's not all that enticing.
Hint: Not that I condone this type of activity as it's costly to other people and businesses ... but, sometimes it's one of those things. But, take a look at other people's landing pages ... especially ones you're sent to through a paid advertisement ... like on Facebook or Adwords. If people are businesses are paying money to send potential leads to a landing page ... there is a decent chance that they're using ... at least ... slightly optimized Headlines and Sub-Headlines. Try and see what they say, find the pain points their using or try and identify any underlying formulas.
The third thing I'd look at is the opt-in form itself. What information are you requesting? Just the email address? Or, are you requesting first name, last name, telephone number, address, and their pet's name? The more fields your form has, the larger the barrier to entry. Only use fields you ABSOLUTELY NEED in order to start the relationship. Don't ever ask for personal information, just because. The simpler, the easier, the better.
The fourth thing I'd look at is the Call-To-Action (CTA) button. This is the "Submit" button for the opt-in form. I'd make sure it stands out (color, dimensions) and the wording matches the offer. For example, "Submit" is a bad CTA. Rarely do people want to "submit" something. BUT, they do want to "Download The Guide!"
Finally, I'd look at the overall look and feel. Is there one objective on the page or are you leading people in many different directions? For example, are you asking for contact information, but also asking for people to share the page? Or to look at some blog articles, or to do anything other than opt-in.
Each landing page should have one specific goal.
I hope that answers your question!