Yes, it can, but as a spider, it will provide very vague - if not barely useful - information.
The description of Intelligence in the rules states:
Intelligence measures mental acuity. accuracy of recall, and the ability to reason.
Telepathy is typically a one way link of communication. You can initiate it and perceive a response.
A beast is a limited Int creature that follows its own instincts.
Speak with Animals doesn’t directly coordinate with the telepathic communication. It confers a magical ability to understand the beast in its own natural forms of communication. This doesn’t mean you can teach it how to program a VCR to stop blinking 12:00.
A Ranger's companion is another reference towards interaction with a beast on a mundane level. Albeit a magical bond, it still is based off physically or verbal interactions. The beast will follow your commands to the best of its ability. So you can’t command the pig to fly, because the pig doesn’t naturally or instinctively know how to.
A beast outside of your 100 ft. range could only give you, at most, the vaguest of details that are barely useful, depending on the beast in question.
Some real-life animals can be trained to respond to situations in a certain manner, as well as actually having the intelligence of a young child and even self-awareness.
A spider is limited to its instincts and its senses. We have limited understanding about what a spider may think, but we have perceived what they can do.
Bugs and insects have inherent amazing abilities, but have very limited cognitive abilities, if capable at all.