The European Union's Artificial Intelligence Act (AI Act), finalized in February 2024, mandates comprehensive transparency and explainability requirements for AI systems to enable effective oversight and safeguard fundamental rights. However, the practical implementation of these requirements faces challenges due to tensions between the need for meaningful explanations and the potential risks to intellectual property and commercial interests of AI providers. This research proposes the Transparency-Explainability Functionality and Tensions (TEFT) framework to systematically analyze the complex interplay of legal, technical, and socio-ethical factors shaping the realization of algorithmic transparency and explainability in the EU context. Through a two-pronged approach combining a focused literature review and an in-depth examination of the AI Act's provisions, we identify key friction points and challenges in operationalizing the right to explanation. The TEFT framework maps the interests and incentives of various stakeholders, including AI providers & deployers, oversight bodies, and affected individuals, while considering their goals, expected benefits, risks, possible negative impacts, and context to algorithmic explainability.